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This game is played with a fair coin and a die. First player flips a coin. If it turns out head(H), the player proceeds with tossing a die. If it turns out tail(T), the player proceeds with flipping a coin for the second time. The player wins if it gets head on the first tossing and 6 on the second or tails on both flips of coin. What is the probability of winning a game?

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    $\begingroup$ Your rules of winning are a bit confusing. Should it be: "The player wins if (1) the first coin toss is H and the subsequent die roll is 6 or (2) both coin flip are tails."? $\endgroup$ – jdods May 3 '16 at 22:52
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First flip and second toss are independent events. So do first flip and second flip in the case that first flip is tail.

So use multiplication:

P(head on the first flip and 6 on the second tossing)=P(head on the first flip)*P(6 on the second tossing)=$\frac{1}{2}*\frac{1}{6}=\frac{1}{12}$

P(tails on both flip)=$\frac{1}{2}*\frac{1}{2}=\frac{1}{4}$

Win the game if either one of the two events happens, so use addition:

P(winning the game)=P(head on the first flip and 6 on the second tossing)+P(tails on both flip)=$\frac{1}{12}+\frac{1}{4}=\frac{1}{3}$

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