Question

A die is rolled. Consider events : \( A = \{1, 2, 5\} \), \( B = \{3, 5\} \), \( C = \{2, 3, 4, 5\} \). Find \( P(A|C) \) and \( P(C|A) \).

Hint:

For discrete finite sample spaces, conditional probability \( P(X|Y) \) is simply the number of outcomes common to both sets divided by the total outcomes in the "given" set.

Answer 1

Concept:
• Sample space for a die: \( S = \{1, 2, 3, 4, 5, 6\} \), \( n(S) = 6 \).
• Conditional Probability: \( P(X|Y) = \frac{P(X \cap Y)}{P(Y)} = \frac{n(X \cap Y)}{n(Y)} \).

Step 1: Find the intersection and individual event counts
Events are: \( A = \{1, 2, 5\} \), \( C = \{2, 3, 4, 5\} \).
Intersection \( A \cap C = \{2, 5\} \).
Counts: \( n(A) = 3 \), \( n(C) = 4 \), \( n(A \cap C) = 2 \).

Step 2: Calculate \( P(A|C) \)
Using the definition of conditional probability:
\[ P(A|C) = \frac{n(A \cap C)}{n(C)} = \frac{2}{4} = \frac{1}{2} \]

Step 3: Calculate \( P(C|A) \)
Using the definition of conditional probability:
\[ P(C|A) = \frac{n(A \cap C)}{n(A)} = \frac{2}{3} \]