Given that S is 7, what is the probability that the first die showed 4 dots?

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BlackTom AI · Accepted Answer

Start by recognizing the scenario: two fair dice sum to S = 7. When conditioning on the sum being 7, only pairs of die results that add up to 7 are possible.
List the all possible ordered pairs that yield a total of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Each pair is equally likely under fair dice assumptions, so there are 6 equally probable outcomes.
Next, identify which of these outcomes have the first die showing 4. Only the pair (4,3) satisfies this condition.
Therefore, the probability is the number of favorable outcomes (1) divided by the total number of outcomes (6), giving 1/6 ≈ 0.1667.
To connect with common intuition: given the sum is 7, the first die is equally likely to be any value from 1 to 6, but not all values occur with equal frequency in the sum-7 constraint; counting shows that only one of the six sum-7 outcomes has a first die of 4, hence 1/6. This matches the decimal 0.1667 when rounded to four decimal places.

COMM_V 190 101-104 2025W1 Final Exam - Practice Questions - Second Half

Given that S is 7, what is the probability that the first die showed 4 dots?

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