The Probability of Poker Hands

Why are Casinos profitable?


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

In n is a positive integer then the factorial of n is defined by: n! = (n)(n-1)(n-2)....(3)(2)(1).

By definition 0!=1


  1. 6!= 6x5x4x3x2x1= 720
  2. 5!= 5x4x3x2x1= 120

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


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:

  1. Task T1: Choose a letter from the alphabet; n1=26.
  2. Task T2: Choose a second letter; n2=25 (no letter is repeated)
  3. Task T3: Choose a third letter; n3=24 (no letter is repeated)
  4. Task T4: Choose a number; n4=10
  5. Task T5: Choose a second number; n5=9 (no number is repeated)
  6. Task T6: Choose a third number; n6=8 (no number is repeated)
  7. 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

A 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:

  1. A pair
  2. Two pairs
  3. 3 of a kind
  4. 4 of a kind
  5. Full House (2 of a kind and 3 of another kind)
  6. Flush
  7. Straight Flush
  8. 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%

Permutations and Combinations - 5 Card Poker Hands

Why are Casinos profitable?

The probabilities of poker games, whether card games or dice games are always very small. We have just calculated the probability of a Royal Flush in a game of poker which turned out to be 0.00154%, which is very minimal. Casinos depend on the fact that probabilities of such games are small. This is how they profit from gambling! You are advised never to gamble.

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