I came across the following problem today.
Flip four coins. For every head, you get $\$1$. You may reflip one coin after the four flips. Calculate the expected returns.
I know that the expected value without the extra flip is $\$2$. However, I am unsure of how to condition on the extra flips. I am tempted to claim that having the reflip simply adds $\$\frac{1}{2}$ to each case with tails since the only thing which affects the reflip is whether there are tails or not, but my gut tells me this is wrong. I am also told the correct returns is $\$\frac{79}{32}$ and I have no idea where this comes from.
Best Answer
Your temptation is right and your gut is wrong. You do get an extra $\frac12$ if you got tails at least once. The probability that you don't have a tail to reflip is $\frac1{16}$, so you get an extra $\frac12\left(1-\frac1{16}\right)=\frac{15}{32}$. This added to the base expectation of $2 = \frac{64}{32}$ gives $\frac{79}{32}$.