# Conditional Probability and Independence - Find the probability that player 1 wins the game.

Suppose that there are ten cards numbered from 1 to 10. The cards are shuffled and the game is played between two players: player 1 and player 2. Player 1 starts the game. Player 1 wins if he selects any card numbered from 1 to 4 both inclusive and the game ends. If player 1 fails to draw the card with desired number, then player 2 draws a card and he wins if any card numbered from 5 to 10 is drawn. If player 2 fails to draw the desired card the game goes back to player 1 and he draws the card again. The players keep on playing the game until one of the player wins. Find the probability that player 1 wins the game.

• are cards replaced once they have been choosen – Manish Kumar Singh Oct 6 '15 at 16:48
• Assuming no replacement, brute force works pretty well, here. If Player 1 wins, it will be on one of his first four draws. There aren't very many terms to calculate. – user3294068 Oct 6 '15 at 16:56
• @ManishKumarSingh with no replacement – chauka khan Oct 6 '15 at 17:07

Suppose cards are replaced after each draw,\begin{align}P(\text{player 1 wins})&=\frac4{10}+(\frac6{10}\frac4{10})\frac4{10}+(\frac6{10}\frac4{10})(\frac6{10}\frac4{10})\frac4{10}+...\\&=\frac4{10}\left(1+0.24+0.24^2+...\right)\\&=\frac4{10}\frac1{1-0.24}\\&=10/19\end{align}
For no replacement,$$P(\text{player 1 wins})=\frac4{10}+\frac6{10}\frac49\frac38+\frac6{10}\frac49\frac58\frac37\frac26+\frac6{10}\frac49\frac58\frac37\frac46\frac25\frac14=37/70$$