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Combinatorics & Probability Section 3.4

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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.

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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, 4 C 3 =4. How many ways to choose the 8s? ANSWER 4 choose 2, 4 C 2 =6. Now multiply 6 and 4 and you get the number of ways of getting Kings full of 8s 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%

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Lets 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 52 C 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 cant be kings so 48 choose 2. Thus we have 4 C 3 =4 and 48 C 2 =1128 Therefore the probability of having a poker hand with exactly 3 kings is =4512/2598960=0.001736

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Lets 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%.

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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 dont 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 dont re count the straight flushes and royal flushes.) Compute the probability and odds of each.

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