Combinatorics & Probability

Section 3.4

Which Counting Technique?

• If the problem involves more than one category, use the Fundamental Principle of Counting.

• Within any one category, if the order of selection is important use Permutations.

• Within any one category, if the order of selection is not important, use Combinations.

A Full House• What is a full house? An example would be three Ks and two 8s.

We would call this Kings full eights. • How many full houses are there when playing 5 card poker?• First think of the example: How many ways to choose 3 kings?

ANSWER 4 choose 3, 4C3=4. • How many ways to choose the 8s? ANSWER 4 choose 2, 4C2=6.• Now multiply 6 and 4 and you get the number of ways of getting

Kings full of 8’s which is 24.• A full house is any three of kind with a pair. So take 24 and

multiply by 13 (13 ranks for the three of a kind) and by 12 (12 ranks for the pair, note you used one rank to make the three of a kind).

• So the number of full house hands is 13x4x12x6=3744. • What is the probability of getting a full house?• ANSWER 3744/2598960=0.00144=0.144%

Let’s Go Further and talk about a three of a kind

• What is the probability of having exactly three Kings in a 5-card poker hand.

• First, how many 5-card poker hands are there? • ANSWER: 52 choose 5 or 52C5= which is 2,598,960• Now how do we figure out a hand that has exactly 3

kings?• ANSWER: There are 4 kings so we choose 3. The other

2 cards can’t be kings so 48 choose 2.• Thus we have 4C3=4 and 48C2=1128• Therefore the probability of having a poker hand with exactly 3 kings is =4512/2598960=0.001736

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Let’s go further

• What is the probability of being dealt a three of a kind. This is a little different from the last problem. Last problem we had a specific three of a kind, so now we can multiply the result by 13 (since there are 13 ranks). So the number of hands that have a three of a kind in them is 58656. Some of these hands are actually full houses. So we should subtract from this result. Which would give 54912. Hence the probability of being dealt a three of a kind (not a full house) is 54912/2598960=0.0211=2.11%.

FLUSH

• Figure out the number of ways you can get a Royal Straight Flush (A,K,Q,J,10 of the same suit) in 5 card poker.

• Figure out how many straight flushes you can get in 5 card poker. (example 8,7,6,5,4 of the same suit and don’t recount the royal flushes.)

• NOTE THIS IS NOT A STRAIGHT Q,K,A,2,3 NO WRAPAROUND.

• Figure out how many flushes in a 5 card poker hand. (Note don’t re count the straight flushes and royal flushes.)

• Compute the probability and odds of each.