Spring 2010 Problem 3: Cheating at poker
You are playing regular five-card stud poker (where the hands are ranked high to low as straight flush, four of a kind, full house, flush, straight, three of a kind, two pair, one pair, and high card). You cheat — sad, but true. For the next hand, you will be giving yourself a full house. Which full house should you give yourself to maximize your chances of winning the hand? (Note: there are multiple answers — I'll take any correct answer. If you have two full houses, ties are broken by first looking at the triple, then the pair.)
Solution
Alright, this one was too hard (sorry, folks!). Only one non-student, Don Dinnerville, got it right, and based on his answer, he plays a lot of poker.
Note that full houses are only beat by higher full houses, fours of a kind, and straight flushes. No matter what full house you pick, you do not alter the total number of possible fours of a kind (there will still be 11). So you want to minimize the total number of full houses higher than your hand plus the number of straight flushes.
If you pick aces for the three of a kind, no other full houses can beat you. Note that there are 40 straight flushes: AKQJ10, KQJ109, QJ1098, J10987, 109876, 98765, 87654, 76543, 65432, and 5432A in four suits. With your aces, you have prevented six straight flushes (two straights in three different suits).
If you pick kings as your pair, you only prevent at most three straight flushes (assuming one king matches none of the aces' suits give two straight flushes plus KQJ109 for the other king). If you pick pairs of nines, eights, sevens, or sixes, you prevent ten straight flushes (five of the above straights in two suits). So you want AAA99, AAA88, AAA77, or AAA66.
But what if you don't choose aces as your three of a kind to block more straight flushes (as Dr. Goadrich and Josh Pena asked)? Say you choose 9H, 9C, 9S, 8S, 8D as your full house. You have blocked five straight flushes in hearts, diamonds, and clubs, and six in spades. But you have allowed many more full houses to beat you (e.g., trip aces and all pairs except nines, trip kings and all pairs except nines, etc...). Going lower with the three of kind introduces more full houses that beat your hand than straight flushes that you prevent. Or that's the idea...