Internal Assessment
Subject: - Mathematics AI HL
Area of exploration: - Probability
Topic: - Probabilities in Casino
Probability is the branch of Mathematics concerning numerical descriptions of
how likely an event is to occur or how likely it is that a proposition is true.
Probability formula is the ratio of number of favorable outcomes to the total
number of possible outcomes. Measures the likelihood of an event in the
following way: -
If P(A) > P(B) then event A is more likely to occur than event B.
For example, the probability of flipping a coin and it being heads is ½, because
there is 1 way of getting a head and the total number of possible outcomes is 2
(a head or tail). We write P(heads) = ½.
Types of Probability: -
There are three Types of Probability
1. Classical:
(equally probable outcomes) Let S=sample space (set of all possible distinct
outcomes). Then the probability of an event =
2. Relative Frequency Definition
The probability of an event in an experiment is the proportion (or fraction) of
times the event occurs in a very long (theoretically infinite) series of
(independent) repetitions of experiment. (e.g. probability of heads=0.4992)
3. Subjective Probability
The probability of an event is a "best guess" by a person making the statement
of the chances that the event will happen. (e.g. 30% chance of rain).
Applications
Probability theory is applied in everyday life in risk assessment and
modeling.
The insurance industry and markets use actuarial science to determine
pricing and make trading decisions.
Meteorologists, for instance, use weather patterns to predict the
probability of rain. In epidemiology.
Probability theory is used to understand the relationship between
exposures and the risk of health effects.
We use probability to predict results of experiment under assumptions.
Compute probability of error larger than given amount. Compute
probability of given departure between prediction and results under
assumption. Decide whether or not assumptions likely realistic.
Gambling mathematics: -
The mathematics of gambling are a collection of probability applications
encountered in games of chance and can be included in game theory. From a
mathematical point of view, the games of chance are experiments generating
various types of aleatory events, the probability of which can be calculated by
using the properties of probability on a finite space of events.
How do casinos use probability?
Dealing cards in blackjack is an experiment that generates events such as the
occurrence of a certain card or value as the first card dealt, obtaining a certain
total of points from the first two cards dealt, exceeding 21 points from the first
three cards dealt, and so on. In card games we encounter many types of
experiments and categories of events. Each type of experiment has its own
sample space.
For example, the experiment of dealing the first card to the first player has as its
sample space the set of all 52 cards (or 104, if played with two decks). The
experiment of dealing the second card to the first player has as its sample space
the set of all 52 cards (or 104), less the first card dealt.
The experiment of dealing the first two cards to the first player has as its sample
space a set of ordered pairs, namely all the 2-size arrangements of cards from
the 52 (or 104). In a game with one player, the event the player is dealt a card of
10 points as the first dealt card is represented by the set of cards {10♠, 10♣,
10♥, 10♦, J♠, J♣, J♥, J♦, Q♠, Q♣, Q♥, Q♦, K♠, K♣, K♥, K♦}.
The event the player is dealt a total of five points from the first two dealt cards
is represented by the set of 2-size combinations of card values {(A, 4), (2, 3)},
which in fact counts 4 x 4 + 4 x 4 = 32 combinations of cards (as value and
symbol).
Another example, a game is played by wagering on the number that would
result from the roll of one die, true odds would be 5 times the amount wagered
since there is a 1/6 probability of any single number appearing. However, the
casino may only pay 4 times the amount wagered for a winning wager.
These are a few examples of gambling events, whose properties of
compounders, exclusiveness and independency are easily observable. These
properties are very important in practical probability calculus.
The complete mathematical model is given by the probability field attached to
the experiment, which is the triple sample space—field of events—probability
function. For any game of chance, the probability model is of the simplest type
—the sample space is finite, the space of events is the set of parts of the sample
space, implicitly finite, too, and the probability function is given by the
definition of probability on a finite space of events.
A probability model starts from an experiment and a mathematical structure
attached to that experiment, namely the space (field) of events. The event is the
main unit probability theory works on.
It is fact that there are more losers in casino games than the winners.
My research will be on the role of probability for the games played in casinos.
