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= 52! (52 5)!5! 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 4 C3 48 C 2 =4512/2598960=0.001736 52 C5 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.