Simple Gambling

Betting Systems
Conclusions

Betting systems: how not to lose your money

gambling

G. Berkolaiko

Department of Mathematics
Texas A&M University

28 April 2007 / Mini Fair, Math Awareness Month 2007

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambling and Games of Chance

Simple games:
flipping a coin (head / tail)
rolling a die (6 sided)
roulette (18 black, 18 red and 2 green)
Player vs casino
Player predicts outcome and agrees on the wager
If prediction is correct, the player gets money, otherwise he
pays.
Examples:
if head, player wins 1$;
Coin:
if tail, player loses 1$.
if 5 or above, win 5$;
Die:
if 4 or below, lose 3$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ —
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ —
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ —
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ —
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ — subfair
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Fair Game

Examples:
Coin tossing with wager 1$ — fair
Roulette red-black, wager 1$ — unfair to player or subfair
Die: 5 or above win 2$, 4 or below lose 1$ — fair
Die: 5 or above win 5$, 4 or below lose 3$ — subfair
sum 8 or above win 20$,
2 dice: —?
sum 7 or below lose 15$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Definition

average = sum of (probability × payment).

(
positive if a win,
payment =
negative if a loss.

# favourable combinations
probability =
# all combinations

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Definition

average = sum of (probability × payment).

(
positive if a win,
payment =
negative if a loss.

# favourable combinations
probability =
# all combinations

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Definition

average = sum of (probability × payment).

(
positive if a win,
payment =
negative if a loss.

# favourable combinations
probability =
# all combinations

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 =0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 =0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 =0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 =0
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Average: Examples

Coin tossing
head probability = 1/2.
tail probability = 1/2.
average = 1 × 12 + (−1) × 1
2 = 0 — fair!
Roulette
#total = 18 + 18 + 2 = 38.
#favourable = 18.
average = 1 × 18 20 2
38 + (−1) × 38 = − 38 < 0 — subfair!
2 dice:
#total combinations = 36.
#combinations with sum 8 or larger = 15.
#combinations with sum 7 or smaller = 21.
average = 20 × 15 21 15
36 + (−15) × 36 = − 36 < 0 — subfair!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Another type of average

Previous average was the average winning in one round.

Another important average is average capital after round N:
Example:
you had 10$ and played a round of coin tossing.
if you won (probability 1/2), you have 11$.
if you lost (probability 1/2), you have 9$.
on average you have 9 × 12 + 11 × 12 = 10 dollars.
Example:
you had 10$ and played a round of roulette.
on average you have 9 × 20 18 36
38 + 10 × 38 = 9 38 = 10 −
2
38 dollars.
You had 10$ and played roulette. What is you average capital
after round 19? 9$.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Problem

Initially you have n dollars.

You play a game with a wager of 1$ in each round.
In each round, the probability to win is p.
p = 1/2 is fair, p < 1/2 is subfair.
You stop when you get to m dollars or lose all your money.
What is the probability to win the day?
Looking from the other angle, what is the probability of you
financial ruin?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Formula

Probability theory grew out of this problem.

And it provides the answer:
ρn −1
(
ρm −1 if ρ 6= 1,
P(n → m) = n
m if ρ = 1,

where
1−p
ρ= .
p

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin : The Formula

Probability theory grew out of this problem.

And it provides the answer:
ρn −1
(
ρm −1 if ρ 6= 1,
P(n → m) = n
m if ρ = 1,

where
1−p
ρ= .
p

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling Basic Setup and Examples
Betting Systems The Concepts of Fair Game and Average
Conclusions Gambler’s Ruin

Gambler’s Ruin: Examples

You start with 900$, your goal is 1000$,

Tossing a coin, probability of success is 0.9.
Playing roulette, P = 0.00003.
You start with 100$, and must raise 20,000$ by dawn,
Tossing a coin, probability to get the money is 0.005.
Playing roulette, it is 0.0000...003$ 911 zeros!!!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Betting Systems: Definition

Assume in each round, possible win = possible loss.

Assume you can bet any amount, if you have the money.
Before each round you decide how much you will bet, based
only on the outcomes of the previous rounds.
Simple examples:
Straight Play: bet 1$ every round.
Chicken Play: bet 1$ every round until the first loss, then stop.
In probability theory, the random variable
“capital after round N”
is called martingale (if the game is fair) and supermartingale
(if the game is subfair).
Before being adopted as a mathematical term, martingale was
used to refer to a particular betting system.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Matringale: An Example of a Betting System

Algorithm:
1 Bet 1$.
2 If you lost, bet 2$.
3 If lost again, bet 4$.
4 And so forth, doubling the stake each round until you win.
5 If you win, go to stage 1.
Analysis:
Suppose you lost 3 times in a row and then won. Your total
win (loss) is −1 − 2 − 4 + 8 = 1.
In general, when a losing streak is followed by a win, the win
pays for the losses and brings you 1$ on top.
Benefit: you can never lose!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Martingale: A Practitioner’s Testimonial

An excerpt from Casanova’s diary, Venice, 1754:

Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]

Alas, several days later,

I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Martingale: A Practitioner’s Testimonial

An excerpt from Casanova’s diary, Venice, 1754:

Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]

Alas, several days later,

I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Martingale: A Practitioner’s Testimonial

An excerpt from Casanova’s diary, Venice, 1754:

Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]

Alas, several days later,

I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

The Martingale: A Practitioner’s Testimonial

An excerpt from Casanova’s diary, Venice, 1754:

Playing the martingale, and doubling my stakes
continuously, I won every day during the remainder
of the carnival. [..] I congratulated myself upon
having increased the treasure of my dear mistress [..]

Alas, several days later,

I still played on the martingale, but with such bad
luck that I was soon left without a sequin. As I
shared my property with my mistress, [..] at her
request I sold all her diamonds, losing what I got for
them; she had now only five hundred sequins by her.
There was no more talk of her escaping from the
convent, for we had nothing to live on!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

You Cannot Reverse the Odds

Problem: you need a lot of money if a losing streak is long.

The martingale is a sure thing only with infinite capital!

Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.

Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.

If the game is subfair, you lose no matter how you bet.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

You Cannot Reverse the Odds

Problem: you need a lot of money if a losing streak is long.

The martingale is a sure thing only with infinite capital!

Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.

Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.

If the game is subfair, you lose no matter how you bet.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

You Cannot Reverse the Odds

Problem: you need a lot of money if a losing streak is long.

The martingale is a sure thing only with infinite capital!

Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.

Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.

If the game is subfair, you lose no matter how you bet.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

You Cannot Reverse the Odds

Problem: you need a lot of money if a losing streak is long.

The martingale is a sure thing only with infinite capital!

Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.

Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.

If the game is subfair, you lose no matter how you bet.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

You Cannot Reverse the Odds

Problem: you need a lot of money if a losing streak is long.

The martingale is a sure thing only with infinite capital!

Theorem
If the game is fair, your average capital remains constant regardless
of the betting system.

Theorem
If the game is subfair, your average capital decreases after every
round regardless of the betting system.

If the game is subfair, you lose no matter how you bet.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Example: Gambler’s Ruin with the Martingale

Are betting systems completely worthless then?

Remember example: you have 900$, you want to reach 1000$.
Suppose the game is fair
Using Straight Play, the probability to reach the target is 0.9.
Using the Martingale, the probability remains the same, 0.9.
Suppose you play roulette
Using Straight Play, the probability is 0.00003.
But with the Martingale, it is 0.783!
This is much better than Straight Play!
And not much worse than in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play System

If the game is subfair, every round we lose some (on average).

To minimize losses — minimize rounds!
Algorithm (Bold Play):
1 Suppose you are within x dollar of your target.
2 If your can afford a bet of x, do it!
3 If your cannot afford, bet all you have got!

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play: How Does it Fare?

In roulette, going from 900$ to 1000$ will be sucessful with

probability 0.88.
Better than Straight Play (0.00003)
or Martingale (0.78)
and not much worse than in a fair game (0.9).
Raising 20,000$ from 100$ at roulette will succeed with
probability 0.003.
Unlikely
but infinitely better than 0.00...03 with Straight Play
and not much worse than 0.005 chance in a fair game.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems: Definition and Examples
Betting Systems
Can Betting Systems Help?
Conclusions

Bold Play is Optimal

Theorem
For every fixed subfair game, starting capital n and the target m,
Bold Play provides the best chances of success.

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems
Conclusions

Conclusions

You cannot reverse the odds.

If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems
Conclusions

Conclusions

You cannot reverse the odds.

If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems
Conclusions

Conclusions

You cannot reverse the odds.

If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?

G. Berkolaiko Betting systems: how not to lose your money gambling

 Simple Gambling
Betting Systems
Conclusions

Conclusions

You cannot reverse the odds.

If you must play and have a target,
Bold Play will maximize your chances of reaching it.
How not to lose your money gambling?

DO NOT GAMBLE!

G. Berkolaiko Betting systems: how not to lose your money gambling