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More heads?


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Player A has one more coin than player B.  Both players throw all of their coins simultaneously and observe the number that come up heads.  Assuming all the coins are fair, what is the probability that A obtains more heads than B?

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Let B have n coins and let A have n+1 coins.

Keep in mind that coin tosses are independent events. Thus, tossing them simultaneously is not material to the outcome.

Suppose A keeps one of his coins in his hand while all the other 2n coins are tossed simultaneously. By symmetry, the expected number of heads tossed by B will equal the expected number of heads tossed by A. Now A tosses his remaining (n+1)st coin. It will be heads with probability of 1/2.

The probability that A obtains more heads than B (obtains) is 1/2.

 

 

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