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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.

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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

  • $m_1 = p_1 \times 1 + (1-p_1)\times(1+s_1)$
  • $m_2 = p_1 \times 0 + (1-p_1)\times(1+s_2)$
  • $s_1 = p_2 \times 0 + (1-p_2)\times(1+m_1)$
  • $s_2 = p_2 \times 1 + (1-p_2)\times(1+m_2)$

You can eliminate $s_1$ and $s_2$ and then solve for $m_1$ and $m_2$

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