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A fair coin is tossed. If a head turns up, a fair die is tossed; if a tail turns up, two fair dice are tossed. What is the probability that the face (or the sum of the faces) showing on the die (or the dice) is equal to 6?

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Take it one step at a time.

  • With probability $\frac12$ you toss a head and then roll one die; the probability that the die comes up $6$ is $\frac16$, so the overall probability of getting your total of $6$ in this way is $\frac12\cdot\frac16=\frac1{12}$.

  • With probability $\frac12$ you toss a head and then roll two dice. What’s the probability of getting a total of $6$ when you roll two dice? Half that is the probability of getting your total of $6$ in this way.

Now just add the two partial results to get the desired probability.

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    $\begingroup$ @Peter: For a little bit, anyway. $\endgroup$ – Brian M. Scott Apr 7 '13 at 3:17
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$P$(throwing a head and rolling a six) = $\frac{1}{2}*\frac{1}{6}$.
$P$(throwing a tails and dice sum to six or six's are showing) = $\frac{1}{2}*\frac{5}{36}$.

The reason it is $\frac{5}{36}$ is this. You have 5 choices for the dice to sum to six: $(1,5),(2,4),(3,3),(4,2),$and $(5,1)$. There are a total of 36 possibilities for the two dice also.
Therefore the answer is $\frac{1}{12}+\frac{5}{72}$ = $\frac{11}{72}$.

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