Flipping Out

Probability Level 3

In front of you are two coins. They are identical in appearance, but one is fair (when flipped, it comes up Heads 50% of the time), while the other comes up Heads 75% of the time. Your task is to determine which one is the unfair coin.

You will only be permitted two flips total. After you choose your first coin and flip it, you can base your decision of which coin to flip second on your results of the first flip. After you perform your second flip, you will be asked to declare which coin you believe to be the fake.

If you always act so as to maximize the likelihood of your eventually correctly identifying the fake coin, what is the probability that you will correctly identify the fake coin?

If the desired probability is written as mn\frac{m}{n}, where mm and nn are positive coprime integers, then enter m+nm+n as your answer.


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