Suppose in a game of 2 player, they will shoot a target in turn. Each shoot is independent of other shoot. The game ends once the target is being hit. The probability of success hitting the target for player $i$ $=p_i,$ where $i=1,2$ . Suppose $m_i,$, where $i=1,2$ are the mean number of shoots taken before the game end for player i. Find $m_1,m_2$. I tried to use theorem of expectation $E(X)=E(E(X|Y))$ but don't know how to apply it.
Suppose $s_i$ are the mean numbers of shoots taken before the game end for player $i$ if the second player is about to take the next shot. Then you have
You can eliminate $s_1$ and $s_2$ and then solve for $m_1$ and $m_2$