I play a game. Suppose I toss 4 fair coins onto a table. For each head I see on the table, I earn a dollar unless I chose to retoss: I get to re-toss just once if I want to, after which I would earn a dollar for each head I see after the retoss. What are my expected earnings if I retoss only if my first toss has 0, 1, or 2 heads?
Here's what I've tried. Let $X$ be the number of heads on the first toss and $Y$ be my final earnings in dollars. By conditioning on $X$, I get
$ E(Y) = E(Y | X = 0, 1, 2)P(X = 0, 1, 2) + E(Y | X = 3)P(X = 3) + E(Y | X = 4)P(X = 4) $
However, how do I find $E(Y | X = 0, 1, 2)$, $E(Y | X = 3)$, and $E(Y | X = 4)$?