# The Probability of Poker Hands

### Why are Casinos profitable?

## Probability

__Definition:__The probability of an event measures the chance for the event to happen. It is always a number between 0 and 1.

The probability of an impossible event is zero, and the probability of a certain event is one.

__Theorem:__ In an experiment, let *n *denote the number of possible outcomes. If *E* is an event of the experiment and *m * is the number of possible outcomes for this event, then the probability of *E* is given by: P(*E*) = *m*/*n*. The percentage for E to happen is 100xP(*E*).

## Factorial Notation

By definition 0!=1

__Examples:__

- 6!= 6x5x4x3x2x1= 720
- 5!= 5x4x3x2x1= 120

__Note:__ The previous examples show that 6!= (6)(5!). In fact we have the following general rule: n!= (n)[(n-1)!]

## Counting

__Theorem 1 (Multiplication Rule):__Suppose there are n1 ways to perform task T1, n2 ways to perform task T2,..., nk ways to perform task Tk, then there are: n1xn2x....nk ways to perform tasks 1, 2,..., k in a sequence.

__Example 1:__ A car license plate consists of 3 letters and 4 numbers. How many license plates can there be if no letter or number is repeated more than once?

__Answer:__ There are 7 tasks to perform:

- Task T1: Choose a letter from the alphabet; n1=26.
- Task T2: Choose a second letter; n2=25 (no letter is repeated)
- Task T3: Choose a third letter; n3=24 (no letter is repeated)
- Task T4: Choose a number; n4=10
- Task T5: Choose a second number; n5=9 (no number is repeated)
- Task T6: Choose a third number; n6=8 (no number is repeated)
- Task T7: Choose a fourth number; n7=7 (no number is repeated)

The total number of license plates is: 26x25x24x10x9x8x7=78624000.

__Theorem 2 (Addition Rule):__ If T1 and T2 are two separate events, and T is the situation where either event T1 or event T2 will occur, then the number of times event T will occur is shown by: n(T) = n(T1)+ n(T2)

__Example 2:__ A jar contains 10 red marbles and another jar contains 12 blue marbles. Let T1 be the event of choosing one red marble and T2 the event of choosing a blue marble. If T is the situation of choosing a red marble or a blue marble then n(T) = 10+12= 22

## Permutation and Combination

__Definition:__A permutation is an arrangement of a set of objects. The order of the arrangement may or may not be important.

__Example:__ The order of letters and numbers in a license plate is important, but the order of cards received in a card game is not important.

__Theorem 1:__ Given a set of n objects and given that r is a positive integer less than or equal to n. The number of permutations of a subset of r objects in which the order is important is given by nPr = n!/(n-r)!

__Example:__ How many 4 digit numbers can we form if the order is important and the digits are not repeated?

__Answer:__ This is like choosing a set of 4 objects from a set of 10 in which the order is important. Thus there are 4P10= 10!/(10-4)! = 10!/6! = 10x9x8x7 = 5040

__Theorem 2:__ Given a set of n objects and given that r is a positive integer less than or equal to n. The number of permutations of a subset of r objects in which the order is not important is given by nCr = n!/r!(n-r)!

__Example:__ In how many ways can 5 cards be chosen from a deck of 52 cards (the order of cards is not important).

__Answer:__ 5C52= 52!/5! x (52-5)! = 2,598,960.

## Poker Hands

__Poker Hand__is the hand consisting of 5 cards chosen from a deck of 52 cards.

The number of poker hands (outcomes) is given by: *n*=52!/5!(52-5)! = 2,598,960.

An event is a special poker hand. Special events/poker hands are:

- A pair
- Two pairs
- 3 of a kind
- 4 of a kind
- Full House (2 of a kind and 3 of another kind)
- Flush
- Straight Flush
- Royal Flush

e.g. A Royal Flush is the hand consisting of an Ace, King, Queen, Jack, and 10, all in the same suit (hearts, diamonds, spades, and clubs). Clearly there are exactly 4 such hands. Therefore, P(Royal Flush)=4/2598960= 0.00000154, OR 0.000154%