2009 AIME I Problems/Problem 3
A coin that comes up heads with probability and tails with probability independently on each flip is flipped times. Suppose that the probability of three heads and five tails is equal to of the probability of five heads and three tails. Let , where and are relatively prime positive integers. Find .
The probability of three heads and five tails is and the probability of five heads and three tails is .
Therefore, the answer is .
We start as shown above. However, when we get to , we square root both sides to get . We can do this because we know that both and are between and , so they are both positive. Now, we have:
Now, we get .
Rewrite it as :
This can be simplified as
This can be factored into
This yields two solutions: (ignored because it would result in ) or
Therefore, the answer is =
Video Solution 2
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