2009-01-17

Darts

Bengt and Sture are in the pub. They are drinking beer, obviously, and also playing darts. Bengt has invented a new kind of darts game. It is best played on a different kind of board, but in the pub they have to make do with the normal one.
The game goes like this: First Bengt throws three darts, hitting a certain combination of fields. Then Sture throws three darts, trying to copy Bengt's throw by hitting the same fields (in any order). If he succeeds, the round is over and no one gets any points. If he fails, Bengt gets to try to copy his own combination. If he succeeds in the first attempt, he gets three points. If he fails, he gets a second attempt; if that succeeds, he gets two points, otherwise he gets no points. Then the round is over. In the next round Sture starts by setting the combination, and then they alternate like that until they have played an even number of rounds and someone has achieved a previously agreed upon score.
(There is also a rule that you are not allowed to use the same combination twice, but that has no impact on this question.)

What makes this game different from other darts games is that you not only need to have good throwing skills, but you also need to be able to assess your throwing skills well. If you start by throwing a really easy combination, the opponent will almost certainly copy it in his only attempt, and if you start with a really difficult combination, you are very unlikely to be able to copy it even in two attempts. So the trick is to know the probability that you will be able to pull off a certain combination, and then to choose a combination that will have the right probability.

  1. Which probability would be the best to have in your first throw? Assume that both players have the same skill level (and that it remains constant, despite the beer).
    In other words: Provided you have already done a first throw (of three darts), and it is such that each of the following throws has a probability p to copy it, what would you want p to be?
  2. When you make your first throw, what probability should you aim for? Assume that if you miss the intended target with any of the darts on your first throw, you will not get any points (because you will hit something that is either too hard or too easy to copy).
  3. What if you make a slightly more complicated assumption - if you miss with a dart, that dart will hit something that is trivial to copy (such as the wall), but you can still hope to make it a challenging throw by hitting something with the other darts. Which probabilities should you aim for with each of the three darts? (It may depend on whether you hit with the previous darts, so there are seven cases in total.)