I was entertaining myself trying to figure out the following: If my son plays soccer in a team of 11 players, and at any point there are 8 kids playing, and 3 in the bench, what are the probabilities that he'll be playing at any point in time, assuming equal playing time for all players.
If I look at it from the point of view of the players left out, I can reason: For the first player to be seated out he has 1/11 chances of being "chosen". If he's not the one out the first time around, he can still be the second player seated with a probability of 1/10. The third player chosen to rest can turn out to be him with a probability of 1/9. The sum of these probabilities (... OR ...) is 0.294.
Now, if I reason with sample spaces there are 11 choose 8 combinations: 165. And the combinations that include him in the 8 playing combo are 10 choose 7: 120. So the probability is 120 / 165 = 0.7272. And the probability of sitting down is, 0.2727.
Why are the results different with either reasoning?