Many are familiar with the birthday probability problem, but it might be new to some of you. Given a group of people chosen at random, what is the probability that two or more of them celebrate their birthday on the same day of the year? To simplify the problem, assume 365 day years and each day as likely as any other day (these assumptions are only approximately correct). The following are some sample probabilities. When I first encountered this one, it seemed counterintuitive to me. P(22) = .475 695 (Less than 50-50) P(23) = .507 297 (Better than even money) P(30) = .706 316 (Odds 7 to 3 in favor) P(40) = .891 232 (Almost 9 to 1 in favor of a hit) If you flip a true coin 4 times, the probability of THHH is the same as the probability of HHHH, both probabilities are equal to 1/16 or 15 to one against a hit. If you flip a coin many times and record the sequence of heads and tails, what is the probability of THHH occurring before HHHH? Answer 15/16 or 15 to one in favor of THHH occurring first. If you flip a coin millions of times and record the number of times that THHH occurs and the number of times that HHHH occurs, which occurs more frequently? HHHH occurs more frequently. The more frequent event is less likely to occur first. Suppose you make a set of three dice (Red, Blue, & Yellow), using the numbers from 1 to 9. Each die has three of the nine numbers, with opposite sides having the same number. The Red die has numbers 1, 6, 8; The Blue die has 3, 5, 7; The Yellow die has 2, 4, 9. You pick one of the three dice and then I pick one of the remaining two. We play a game by rolling the pair of dice chosen. If my die has a higher number than yours, I win a dollar, and vice versa. If you pick first, I will always win money in the long run. I expect to win 5 times for every 4 times you win. The Red die beats the Blue die with 5 to 4 odds or a 5/9 probability. The Blue die beats the Yellow die with 5 to 4 odds or a 5/9 probability. The Yellow die beats the Red die with 5 to 4 odds or a 5/9 probability. Is likely to win is not transitive like greater than. Gloria and I just returned from a delightful dinner which included 1.75 liters of wine and several cups of coffee heavily laced with Irish Cream Liquor (highly recommended). If I were more sober, I think I could remember some other probability anomalies. There are many of them, because probability is often counterintuitive.