Answer to the Coin Problem

Solution to the Coin Problem

Because the coins are fair, each one has a probability of 1/2 of showing HEADS or TAILS, and they are independent of each other. Therefore, there are four equally likely outcomes of Joe's 2-coin toss:

TT
TH
HT
HH

The question is what is the probability that both coins are heads, given that at least one of the coins is heads?

Given no information, we know that the probability of HH would be 1/4, since HH is one of four equally likely outcomes.

However, we now know that Joe saw at least one H, so he did not see TT. Therefore, there are three equally likely outcomes, TH, HT, and HH. Thus, the probability of HH is 1 out of 3 or 1/3 = .333.

Some people confuse this question for a very similar question: Given the FIRST COIN is HEADS, what is the probability that the SECOND coin is HEADS. The answer to that question is 1/2, since this information would rule out TT and TH, leaving two equally likely outcomes, HT and HH. But Joe wouldn't tell us which coin was Heads, so his information only rules out TT.

To review, the answer is 1/3, which corresponds to 33%.

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