Flip four coins. For every head, you get $\$1$. You may reflip at most two coins after the four flips. Calculate the expected returns.

Question

I am modifying a question asked in MSE here.

Flip four coins. For every head, you get $\$1$. You may reflip at most two coins after the four flips. Calculate the expected returns.

In this case, would the answer be the same as reflipping one coin, which is $$\frac{79}{32}?$$

Answer 1

This seems similar to your earlier question. Without reflips your expected gain is 2. You have 1/4 chance of flipping just one T, in which case you reflip one coin, and so your expected gain increases by $\frac{1}{4}\times\frac{1}{2}=\frac{1}{8}$. You have 11/16 chance of getting at least two T, in which case you reflip two coins, and your expected gain increases by 11/16. So total expected gain $$2+\frac{13}{16}=\frac{45}{16}=\frac{90}{32}$$