The Probability of Poker Hands
Why are Casinos profitable?
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).
By definition 0!=1
- 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)!]
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
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.
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)
- 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%