Counting & Probability

The Math Behind Playing Poker

So, What is Counting?

Counting is a part of everyday life (eg, data coding, arraging possibilities, etc), and an efficient way of counting is necessary to handle large masses of statistical data (eg, the number of production runs on a given machine in 24 hours, etc.), and for an understanding of probability.

For example, what is total number of possible outcomes when a pair of coins is tossed?

The events are described as:

E1 = toss first coin (2 outcomes, so n(E1) = 2.)

E2 = toss second coin (2 outcomes, so n(E2) = 2.)

They are independent, since neither toss affects the outcome of the other toss.

So n(E) = n(E1) × n(E2) = 2 × 2 = 4

What About Probability?

Probabillity was once a measure of chance. It used to be called "the game of chance or hazard". Later on the theory of probability was born which expanded into new domains (for example, Statistics, medicine, insurance companes, etc).

Experiment: This is any process of observation or procedure that:

(1) Can be repeated (theoretically) an infinite number of times; and

(2) Has a well-defined set of possible outcomes.

Sample space: This is the set of all possible outcomes of an experiment.

Event: This is a subset of the sample space of an experiment.

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Factorial Notation

Factorial Notation is a simple way of writing the product of all the positive whole numbers up to a given number. n factorial is defined as the product of all the integers from 1 to n (the order of multiplying does not matter) . We write "n factorial" with an exclamation: n!

For example, 5!=5x4x3x2x1=120

Permutations (Ordered Arrangements)

Permutations (Ordered Arrangements) is an arrangement of a set of objects and the order that we arrange the objects in is important. The formula of premutation is: nPr = n! / (n-r)!

For example, In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf?

This is asking for the number of permutations, since we don't want repetitions. The number of ways is: 5P3 = (5!) / (5-3)! = 60

How Can We Use Counting & Probability to Play Poker?

In the standard game of poker, each player gets 5 cards and places a bet, hoping his cards are "better" than the other players' hands. The game is played with a pack containing 52 cards in 4 suits, each suit consiting of 13 cards. The number of different possible poker hands is found by counting the number of ways that 5 cards can be selected from 52 cards, where the order is not important. Card counting while at a Blackjack table is the only way you can get a long term, statistical advantage over the casino, or you can depend on luck and see where that takes you!

Poker Counting odds and outs

The Mathmatical World Of Poker

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Jad Choucair