Posted by Joel on October 13, 2002 at 14:42:14:
: A,B,C are evenly matched tennis players. Initially A and B play a set, and the winner then plays C. This continues, with the winner always playing the waiting player, until one of the players has won two sets in a row. That player is then declared the overall winner. What is the probability that A is the overall winner?
I guess it IS a trick question, and the trick is that although all are evenly matched, A and B, by being in the first round, have an advantage over C.
I don't know how to compute the probability but I believe it is a sum of the form 1/4 + 1/16 + 1/32 + 1/128 + 1/256 + 1/1024 + 1/2048 ... which seems to be approaching .375. (1/64 and 1/512 are intentionally omitted.
Here is my "attempt" at an upside-down tree diagram of the possible outcomes, which (if the board displays it as I am hoping) may show how I am getting to that.
AB
AC BC
**A** CB --BA-- CA 1/4--C-- BA -CB-- AB
--B-- AC **A** BC 1/16
**A** CB --B-- CA 1/32
--C-- BA --C-- AB
--B-- AC **A** B 1/128
**A** CB --B-- AC 1/256