At least one six, is the sum big?

Question

You roll two fair dice. Someone who can see them tells you truthfully that at least one die shows a 6. Given that, what is the probability that the two dice sum to 10 or more?

Accepted Answer

Restrict attention to outcomes consistent with the clue: at least one 6. Listing the ordered outcomes, (6,1),(6,2),(6,3),(6,4),(6,5),(6,6),(1,6),(2,6),(3,6),(4,6),(5,6). That is 11 outcomes (six with the first die a 6, six with the second a 6, minus the doubly counted (6,6)). Now, which of these 11 have a sum of at least 10? Check each: (6,4)=10,\quad (6,5)=11,\quad (6,6)=12,\quad (4,6)=10,\quad (5,6)=11. That is 5 favorable outcomes. Therefore P(	ext{sum} \ge 10 \mid 	ext{at least one six}) = \frac{5}{11} \approx 0.455. Notice the conditioning matters a lot: without the clue, the chance of…

At least one six, is the sum big?

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