Alex flips a fair coin 20 times. Bert spins a fair coin 21
times. Bert wins if he gets more heads than Alex, else Alex wins.
Note that Alex wins if there is a tie. What is the probability that
Bert wins?
Suppose Alex gets a heads and Bert gets b heads. Then, Alex gets 20-a tails and Bert gets 21-b tails. Take any case where Alex wins. Then, a>=b. Now, change heads to tails and tails to heads. Then, Alex gets 20-a heads and Bert gets 21-b heads. Then, 20-a<=20-b<21-b. Therefore, Bert wins.
Take any case where Bert wins. Then, a<b. Now, change heads to tails and tails to heads. Then, 20-a>20-b, so 20-a>=21-b. Therefore, Alex wins.
If you change heads and tails when Alex wins, then Bert wins. If you change heads and tails when Bert wins, then Alex wins. Therefore, there is a 1-1 correspondence between the times that Alex wins and the times that Bert wins. Then, Alex and Bert each win with probability 1/2.
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Posted by Math Man
on 2018-06-29 21:16:25 |
Solution
