Help Team A Win!
Five members of team A are captured by five members of team B while playing tag. The children of team A, Alexandra, Ben, Claire, David, and Erica, are forced to play a game.
THE GAME: Each one of the five members of team B will stand in front of each member of team A and play “heads or tails” with a fair coin. If a member of team B wins, the member of team A will have to quit playing the game and the member of team B will stay. If the team A member wins, the member of team A will stay there and the member of team B will have to quit playing the game.
After the first round, the remaining members of team B will pick new members of team A randomly, and play the next round. If there are more members of team A than members of team B, remaining members of team A, who are not picked, will be safe (i.e. not be kicked out) for that round. If there are more members of team B than members of team A, any leftover members of team B who don’t have someone to play the game with, will also be safe.
The game will continue until all members of one team have to quit playing.
What is the probability that team A will win the game? In order for team A to win, at least one member of team A needs to be still playing the game by the time all members of team B are out of the game.
Hint: Do not try to solve the problem.
The answer is 0.5.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
The solution I was thinking of when I wrote this problem is simple. If you take a close look at the problem, you realize that team A and team B have a same probability of winning. What is the difference between a team B member randomly picking a team A member, and a team A member randomly picking a team B member? None. You can replace every "team A" with "team B", "team B" with "team A", and it would still be the exact same problem. You know that either team A or team B wins with equal probability. P ( A ) = P ( B ) P ( A ) + P ( B ) = 1 2 × P ( A ) = 1 P ( A ) = 2 1 P ( A ) = 0 . 5