5 Heads and Tails
Dan, Sam, Dimitri, Angie and Joe are playing a game in which each one has a fair coin and everyone tosses it at the same time. If, in the result, there are more heads than tails, the owners of the tail coins lose. If there are more tails than heads, the owners of the head coins lose. If the 5 coins are all heads, or all tails, they repeat the process until someone loses (or some of them).
If the probability that Dan does not lose can be expressed as , where and are coprime positive integers, find
The answer is 5.
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1 solution
There are 2 5 or 32 equiprobable ways the coins can land, 30 of them determining winners. If Dan wins with 3 others, the loser can be any of four people and the winning side can be heads or tails, so there are 8 ways for that to happen. If he wins with 2 others, the other two winners can be any pair of the other 4 (there are ( 2 4 ) , or 6 pairs) and the winning side can be heads or tails, for 12 ways. So the probability Dan wins is 20/30 or 2/3, yielding 5 .