Suppose 2 dice are tossed. If you roll doubles (2 dice have same value), roll again. Continue until the third role. What's the expected total sum?
Let $X_i$ be sum on the $i^{th}$ roll and $D_i$ be rolling doubles on the $i^{th}$ roll.
So far I have:
$E[total] = E[X_1 + X_2|D_1]*P(D_1) + E[X_1|\overline{D_1}]*P(\overline{D_1})$ {basically tower property - ignoring $X_3$ for now}
Now I need $E[X_1+ X_2|D_1]$. The other terms, in order, are 1/6, 3.5, and 5/6.
$E[X_1 + X_2|D_1]$ = $E[E[X_1 + X_2|D_1]|D_2]*P(D_2) + E[E[X_1 + X_2|D_1]|\overline{D_2}]*P(\overline{D_2})$ {tower property again}
My question: Is it correct to say $E[E[X_1 + X_2|D_1]|D_2] = E[X_1 + X_2 |D_1 \cap D_2] $ and $E[E[X_1 + X_2 |D_1]|\overline{D_2}] = E[X_1 + X_2|D_1 \cap \overline{D_2}]$? Can you show why? Otherwise I'm stuck!