Poker Math

Casino poker games are played table stakes, which means a player may bet only with the chips (or

money) he has on the table during a hand. If a player runs out of chips when calling or betting, he

cannot add more until the hand is over, and must go all-in to stay in the hand. When a player goes all-in,

all subsequent wagers by other players go into a separate side pot in which the all-in player has no

interest – he may win the main pot, to which he contributed, but may not win the side pot even if his

hand is the best.

The limits, or absence of limits, on how much a player may bet and raise will dramatically affect the

game dynamics, including players’ decisions and strategies, and the relative balance of luck versus skill in

the game.

In a fixed-limit game, no bet or raise may exceed a specified amount. This amount usually varies with

the betting round, with later rounds allowing higher bets and raises than early rounds. In a $5−$10 fixed-

limit game, for example, players may bet or raise exactly $5 in early rounds and exactly $10 in later

rounds.

Spread limit games are like fixed limit but allow any bet between the two amounts at any time. Thus, in

a $10−$20 spread-limit game, bettors may make wagers of any amount between $10 and $20 at any

time, with the provision that raises must be at least equal to the preceding bet. In pot-limit games, bets

or raises are limited only by the amount of money in the pot at the time the wager is made. In no-limit

games, a player may bet or raise any amount he has in front of him (table stakes limit betting in a hand

to the chips and money on the table). Pot limit and no limit formats are used only for more serious

games (no-limit is used in the World Series of Poker, the premier high-stakes tournament). In most limit

games, a bet and either three or four raises per betting round are permitted.
The probabilities of various poker hands (given in the table above) are determined using the usual

probability rules and methods of counting, including combinations and permutations.

In Hold ‘Em, each player is dealt two cards face down, and then a total of five community cards are dealt

face up in the center of the table. Each player uses the five community cards in combination with his

two-hole cards to form the best five-card hand. After the first two cards are dealt to each player there

is a round of betting. After the first round of betting, the first three community cards (the flop) are

exposed, followed by a second round of betting. After the second betting round, the fourth community

card (the turn) is exposed, followed by another round of betting, and then the fifth and final community

card (the river) is exposed, followed by a final round of betting. Each betting round begins with the first

active player to the left of the dealer (or in a game dealt by a house dealer, the first active player to the

left of the button used to indicate dealer position). Suppose a player has four cards to flush after the

flop. Consider the following three questions:

(a) What is the probability he will make the flush on the turn?

(b) If he does not make the flush with the turn card, what is the probability he will make it on the river?

(c) What is the probability he will make the flush on either the turn or river?

Suppose the player holds two spades and two of the three flop cards are spades. Since the player has

seen five cards – his two-hole cards and the three flop cards – there are 47 remaining unseen cards, of

which nine are spades (i.e., there are nine outs, or cards that will complete the flush). Thus the

probability he will make his flush on the turn card is 9/47 = .191, for odds against of 38 to 9, or about 4.2

to 1, answering question (a). To answer (b), note that if he does not make the flush on the turn, there

are still 9 spades left in the 46 remaining cards, so the probability he makes it on the river is 9/46 = .196,
for odds against of 37 to 9, or 4.1 to 1. To answer (c), first compute the probability he does not make the

flush on the turn or river, in Hold ‘Em game, two blind bets are posted before the cards are dealt – a

small blind by the player to the dealer’s immediate left and a large blind by the next player to the left of

the small blind. A blind is a forced bet made before the player sees his cards used to start the pot and

stimulate action. The small blind is usually equal to one-half the amount of the big blind. Since the deal

rotates around the table (even in a casino where the dealer is not a player, a button used to signify the

nominal dealer rotates after each hand), all players participate equally in the posting of any forced blind

bets. We will refrain from discussing further details regarding betting amounts and structure as it is not

necessary for this example. Because the first two players to the left of the dealer (or button) have

already acted by putting in blind bets, the player one to the left of the big blind is the first with any

choices (to call, raise, or fold in the first round of betting) on the pre-flop betting round.

The probability of making the flush on either the turn or river is 35%, for odds against of 1.86 to 1. This

last calculation illustrates how some probabilities can be easier to determine by first computing the

probability of the opposite (complement), then subtracting the result from one. Note that the

probability that the flush is made with one card to come depends on whether we look at making the

flush on the turn card or the river card (having not made the flush on the turn). In the example above,

the former probability is .191; the latter is .196.126 The following tables show probabilities and odds for

making hands with a given number of outs (cards that will make the desired hand), with the “one card

to come” probabilities and odds computed assuming that one card is the final river card.

Examples of games in which rule variations can affect the odds are blackjack and craps. In blackjack, the

dealer hitting a soft seventeen increases the house advantage 0.2% compared to a comparable game

where the dealer must stand on soft seventeen. The number of decks used, no soft doubling, and no re-

splitting of pairs are other examples of rule variations in blackjack that affect the overall price of the

game to the player.

taken. A player who bets the pass line and takes single odds is at a 0.85% disadvantage, but only a 0.61%

with double odds, and 0.47% with triple odds. This means for every $100 wagered on the pass line with

single odds ($50 pass line and $50 odds), on the average, the player will pay a price of about $0.85.

while $100 bet on the pass line with triple odds ($25 pass line and $75 odds) will cost about $0.47. A

casino allowing triple odds offers a better priced craps game than one that permits only single odds.

Some casinos have offered as high as 100X odds – a player taking full 100X odds will face only a 0.02%

house advantage on the combined pass line (or come) and odds wagers.

Early slot machines were mechanical and the odds of winning depended on the number of reels,

number of stops on each reel, and payouts for the winning combinations. For example, if a slot machine

had 3 reels with 20 stops each, and each stop had a different symbol, there would be (20 x 20 x 20) =

8,000 combinations. If 30% of these combinations were winners with a combined total payout of 7,500

coins, the hold percentage (house advantage) would be (8,000 – 7,500)/8,000 or 6.25%. The average

price of playing this machine would be $6.25 for every $100 bet. Casino operators could adjust this price

by changing the number of winning combinations, changing the symbol configurations on the reels, or

changing the payout for winning combinations. With today’s microchip-controlled slots, the odds of

winning can be adjusted merely by altering the computer program that runs the gaming device.

A factor that is not in the casino’s control that affects game price is player skill in those games involving

both chance and skill. In a typical six-deck game, for example, the average blackjack player gives about a

2% edge to the house, but a basic strategy player is at a 0.5% disadvantage. Game speed, or number of

decisions per hour, affects the cost per hour to the player (and the expected casino win per hour),

although it does not alter the basic price per unit wagered (i.e., the house advantage). In roulette, for

example, a $5-per-spin player betting at a double-zero table making 40 spins per hour can expect to pay
($5 x 40 x .0526) = $10.52 per hour. At a table completing 60 spins each hour, the same player would

spend an average of $15.78 per hour. The basic price in both cases is 5.26% of the amount wagered.

Similarly, at a base price of 1.15%, a $100-per-hand baccarat player will pay, on average, about $92 per

hour if playing 80 hands per hour, but it will cost this same player $161 per hour if dealt 140 hands per

hour.

Because pricing decisions in the casino require mathematical calculation, either a failure to make such

calculations or errors in doing so can have negative consequences. One casino wanted to increase their

baccarat play and so lowered the commission on winning banker bets to 2%. It was not long before the

players made them pay for this mistake. As the following expected value calculation shows, this 2%

commission gives the player a 0.32% advantage:

EV = (+.98) (0.4462) + (−1) (0.4586) = 0.0032.

A player betting $100 per hand at 60 hands per hour can expect to win $750 per hour. A $500 bettor at

this rate would take in $3,750 per hour. An Illinois riverboat lost $200,000 in one day with a “2 to 1

Tuesdays” promotion in which blackjack naturals paid 2 to 1 (instead of the usual 3 to 2). A similar

promotion with related results occurred at a Las Vegas Strip casino. Without other compensating rule

changes, paying naturals 2 to 1 can increase the player expectation enough to give the player about a

2% advantage over the house.