Flipping until you've seen both heads and tails

Question

You flip a fair coin repeatedly and stop the moment you have seen both a head and a tail at least once. What is the expected number of flips until you stop?

Accepted Answer

The first flip always shows one of the two faces, that costs 1 flip and can never complete the goal by itself. After it, you just need to see the other face for the first time. Each subsequent flip matches the missing face with probability 	frac12, independently. The number of flips to get that first "opposite" outcome is geometric with success probability 	frac12, so its expectation is \frac{1}{1/2} = 2. Add the guaranteed first flip: E[	ext{flips}] = 1 + 2 = 3. Takeaway. Break a waiting problem into stages: a certain first step, then a geometric wait for the remaining condition. A geomet…

Flipping until you've seen both heads and tails

Your answer

More Probability questions