Coin flip problem. I'm given the probability distribution of how many times a coin is heads out of 5 flips. From this, I need to determine if the following deal is worth it and, if not, how much it costs using variance:
After flipping the coin 20 times I need to pay my friend the square of the total number of heads results. But, no matter what, they have to pay me $15.
The Attempt at a Solution
I got the expectation value from the distribution to be exactly 2. Meaning, out of 5 flips two results can be expected to be heads. This means I have a 2/5 probability of heads (40%). Out of 20 flips this means 8 are likely to be heads and since 64>15 this is a terrible deal. So immediately I know he'll have to pay around $49 dollars to his friend.
My issue is with variance. The problem hints that the variance of any variable is the E(X2)-E(X)2 and that I already have a simple formula for variance of the probability distribution. I calculated the variance of the expectation using the above equation and got 3.2. But I really don't know what to do with this or what other variance he is speaking about. Can anyone point me in the right direction?