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Dice sum news

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$2$ fair $6$ -sided dice are rolled. A contract pays, in dollars, the sum of the dice. You learn that at least one die shows $6$ . What is the new fair value in dollars? Give an exact fraction.

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Hint

List the outcomes that remain possible.

Solution

The remaining outcomes are the $11$ pairs with at least one $6$ , all equally likely.
Their sums add up to $2 \times (7 + 8 + 9 + 10 + 11) + 12 = 102$ .
The new fair value is $\frac{102}{11}$ .

Answer: $\frac{102}{11}$

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