To find the conditional probability that at least one 5 appears given that the sum of the numbers on two dice is 8 , we need to calculate:
1. The probability of the sum being 8 .
2. The probability of at least one die showing a 5 and the sum being 8 .
3. The conditional probability of at least one die showing a 5 given that the sum is 8 .
The total number of outcomes when a die is thrown twice is
6×6=36.
Step 1: Find the probability that the sum is 8.
The possible outcomes that result in a sum of 8 are:
(2,6)(3,5)(4,4)(5,3)(6,2)There are 5 such outcomes.
Thus, the probability of the sum being 8 is
‌.
Step 2: Find the probability of at least one die showing 5 and the sum being 8.
The favorable outcomes where at least one die shows 5 and the sum is 8 are:
There are 2 such outcomes.
Thus, the probability of at least one die showing 5 and the sum being 8 is
‌=‌.
Step 3: Calculate the conditional probability.
The conditional probability that at least one die is 5 given the sum is 8 is the ratio of the number of favorable outcomes to the number of outcomes where the sum is 8 :
The correct answer is therefore Option B :
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