I have been playing a lot of Yahtzee lately with Rianna, our 8-year-old. If you've been exposed, you know that you play the game with five dice, and that the players in turn get to roll up to three times, setting aside after each roll the results they want to keep. A "Yahtzee" is the same result on all five dice. A pretty big part of the game is getting to three of a kind for all six numbers within the thirteen rounds that make a game. Getting only two of a kind, especially for the higher numbers, will kill your score. The first roll of a turn frequently yields two of one number together with three "singletons." Rianna and I feel that it is an extraordinary run of bad luck not to pick up the needed third on either your second or third roll, yet it happens with sufficient regularity that I began to wonder just what the probability is. The solution must be given by (5/6)6: the 5/6 representing the probablity that a single throw does not yield the desired result, and the exponent standing for the six (three dice times two rolls) straight undesired results. This number turns out to be just about exactly 1/3. In other words, the chance of failing to "triple up" after an initial pair is about the same as the chance of a major league batting champion getting a hit in his next at-bat. He's more likely to make an out, but in a game in which he comes to bat several times you have to expect that he'll get at least one hit. In fact, if he bats four times the chance of four outs is (2/3)4, which is less than 1/5.
Speaking of gaming and probability, isn't it sometimes in the interests of a football team to allow an immediate touchdown? In today's Vikings game, for instance, Denver intercepted a Minnesota pass deep in Minnesota territory with the score tied and around 90 seconds left. To me it seems obvious that the Vikings should have let Denver score a touchdown on the first play. Instead, Denver ran a few plays, the last one being a quarterback keeper to position the ball immediately in front of the uprights, and kicked a winning field goal on the last play of the game. If after the interception Minnesota had let Denver score an immediate touchdown, there would have been more than a minute in which to score a tying touchdown. Isn't there a better chance of that than there is of the other team missing a chip-shot field goal? But I've never seen a team pursue this strategy. The commentators in today's game didn't mention it as a possibility. It's almost as if it's against the ethics, but I don't see why doing what has to be done to increase the chances of victory should be frowned upon. A team trying to win should pursue the optimal strategy.
