Question

A pair of standard unbiased six-sided dice are rolled simultaneously. Given that the sum of the numbers showing is exactly 6, what is the conditional probability that one of the dice shows the number 2?

• A. \( \frac{2}{5} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{6} \)
• D. \( \frac{2}{36} \)

Hint:

For conditional probability problems, avoid using the default sample space of 36 as your denominator. Reducing your sample space immediately based on the given condition phrase makes counting favorable pairs much simpler and less prone to errors.

Answer 1

The Correct Option is A

Concept: Conditional probability measures the likelihood of an event occurring given that a prior condition has already taken place. We can solve this directly by restricting our sample space denominator to only include outcomes that satisfy the given condition: \[ P(A|B) = \frac{n(A \cap B)}{n(B)} \]

Step 1: Determine the size of the restricted condition sample space \( n(B) \).
The condition states that the sum of the numbers showing on the two dice must equal exactly 6. List all possible coordinate pairs that satisfy this sum requirement: \[ B = \{(1,5), \, (2,4), \, (3,3), \, (4,2), \, (5,1)\} \] Counting these outcomes gives our restricted sample space size: \[ n(B) = 5 \]

Step 2: Count the outcomes that satisfy both conditions simultaneously, \( n(A \cap B) \).
Identify which of the pairs in our restricted sample space contain the number 2 on at least one of the dice: \[ A \cap B = \{(2,4), \, (4,2)\} \] Counting these favorable outcomes yields: \[ n(A \cap B) = 2 \]

Step 3: Evaluate the conditional probability ratio.
Divide the number of favorable outcomes by the size of our restricted sample space: \[ P(A|B) = \frac{n(A \cap B)}{n(B)} = \frac{2}{5} \]