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Dice with a partial clue

ProbabilityMedium

$2$ fair $6$ -sided dice are rolled independently. You are told that at least one of them shows a $6$ . What is the probability that the sum is at least $10$ ?

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Hint

Count outcomes in the reduced sample space instead of using the original $36$ .

Solution

Outcomes with at least one $6$ : $36 - 25 = 11$ , all equally likely.
Among them, the sum is at least $10$ for $(6, 4)$ , $(4, 6)$ , $(6, 5)$ , $(5, 6)$ and $(6, 6)$ .
That gives $5$ favourable outcomes out of $11$ .
The probability is $\frac{5}{11}$ .

Answer: $\frac{5}{11}$

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