Multiple choice

A die is thrown twice and the sum of the numbers appearing is observed to be $7$. What is the conditional probability that the number $2$ has appeared at least once?

  1. $\cfrac{1}{6}$
  2. $\cfrac{1}{3}$
  3. $\cfrac{2}{7}$
  4. $\cfrac{3}{5}$
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B Correct answer
Explanation

Sum is 7: {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}. Total 6 outcomes. Outcomes with at least one 2: {(2,5), (5,2)}. Probability = 2/6 = 1/3.

AI explanation

The sample space for a sum of 7 contains exactly 6 pairs: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). The favorable pairs containing the number 2 at least once are (2,5) and (5,2), providing 2 favorable outcomes. By the definition of conditional probability, we divide the favorable outcomes by the total reduced sample space outcomes: 2 divided by 6. The result is 1/3.

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