Question:
A die is rolled. Consider events :
\( A = \{1, 2, 5\}, B = \{3, 5\}, C = \{2, 3, 4, 5\} \)
and hence find :
(i) \( P(A|C) \) and \( P(C|A) \)
(ii) \( P(A \cap B | C) \) and \( P(A \cup B | C) \)
A die is rolled. Consider events :
\( A = \{1, 2, 5\}, B = \{3, 5\}, C = \{2, 3, 4, 5\} \)
and hence find :
(i) \( P(A|C) \) and \( P(C|A) \)
(ii) \( P(A \cap B | C) \) and \( P(A \cup B | C) \)
Show Hint
Conditional probability effectively restricts your sample space to the 'given' event.
To find \( P(X|C) \), simply look at set \( C \) and count how many elements of \( X \) it contains, then divide by \( n(C) \).
To find \( P(X|C) \), simply look at set \( C \) and count how many elements of \( X \) it contains, then divide by \( n(C) \).
Updated On: Sep 10, 2026
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Solution and Explanation
Concept:
• Sample space for a single die roll: \( S = \{1, 2, 3, 4, 5, 6\} \).
• Classical definition of probability: \( P(E) = \frac{n(E)}{n(S)} \).
• Conditional Probability: \( P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{n(E \cap F)}{n(F)} \).
• Sets: \( A \cap B \) is the intersection (common elements) and \( A \cup B \) is the union (all elements from both).
Step 1: Evaluate the basic probabilities of the events
From the sample space \( S \), \( n(S) = 6 \). The given events are: \( A = \{1, 2, 5\} \Rightarrow n(A) = 3 \) \( B = \{3, 5\} \Rightarrow n(B) = 2 \) \( C = \{2, 3, 4, 5\} \Rightarrow n(C) = 4 \)
Step 2: Solve part (i): Find \( P(A|C) \) and \( P(C|A) \)
First, find the intersection \( A \cap C \): \( A \cap C = \{1, 2, 5\} \cap \{2, 3, 4, 5\} = \{2, 5\} \Rightarrow n(A \cap C) = 2 \). \[ P(A|C) = \frac{n(A \cap C)}{n(C)} = \frac{2}{4} = \frac{1}{2} \] \[ P(C|A) = \frac{n(C \cap A)}{n(A)} = \frac{2}{3} \]
Step 3: Solve part (ii): Find \( P(A \cap B | C) \)
First, find \( A \cap B \): \( A \cap B = \{1, 2, 5\} \cap \{3, 5\} = \{5\} \). Now, find the intersection with \( C \): \( (A \cap B) \cap C = \{5\} \cap \{2, 3, 4, 5\} = \{5\} \Rightarrow n((A \cap B) \cap C) = 1 \). \[ P(A \cap B | C) = \frac{n((A \cap B) \cap C)}{n(C)} = \frac{1}{4} \]
Step 4: Solve part (ii): Find \( P(A \cup B | C) \)
First, find \( A \cup B \): \( A \cup B = \{1, 2, 5, 3\} = \{1, 2, 3, 5\} \). Now, find the intersection with \( C \): \( (A \cup B) \cap C = \{1, 2, 3, 5\} \cap \{2, 3, 4, 5\} = \{2, 3, 5\} \Rightarrow n((A \cup B) \cap C) = 3 \). \[ P(A \cup B | C) = \frac{n((A \cup B) \cap C)}{n(C)} = \frac{3}{4} \]
• Sample space for a single die roll: \( S = \{1, 2, 3, 4, 5, 6\} \).
• Classical definition of probability: \( P(E) = \frac{n(E)}{n(S)} \).
• Conditional Probability: \( P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{n(E \cap F)}{n(F)} \).
• Sets: \( A \cap B \) is the intersection (common elements) and \( A \cup B \) is the union (all elements from both).
Step 1: Evaluate the basic probabilities of the events
From the sample space \( S \), \( n(S) = 6 \). The given events are: \( A = \{1, 2, 5\} \Rightarrow n(A) = 3 \) \( B = \{3, 5\} \Rightarrow n(B) = 2 \) \( C = \{2, 3, 4, 5\} \Rightarrow n(C) = 4 \)
Step 2: Solve part (i): Find \( P(A|C) \) and \( P(C|A) \)
First, find the intersection \( A \cap C \): \( A \cap C = \{1, 2, 5\} \cap \{2, 3, 4, 5\} = \{2, 5\} \Rightarrow n(A \cap C) = 2 \). \[ P(A|C) = \frac{n(A \cap C)}{n(C)} = \frac{2}{4} = \frac{1}{2} \] \[ P(C|A) = \frac{n(C \cap A)}{n(A)} = \frac{2}{3} \]
Step 3: Solve part (ii): Find \( P(A \cap B | C) \)
First, find \( A \cap B \): \( A \cap B = \{1, 2, 5\} \cap \{3, 5\} = \{5\} \). Now, find the intersection with \( C \): \( (A \cap B) \cap C = \{5\} \cap \{2, 3, 4, 5\} = \{5\} \Rightarrow n((A \cap B) \cap C) = 1 \). \[ P(A \cap B | C) = \frac{n((A \cap B) \cap C)}{n(C)} = \frac{1}{4} \]
Step 4: Solve part (ii): Find \( P(A \cup B | C) \)
First, find \( A \cup B \): \( A \cup B = \{1, 2, 5, 3\} = \{1, 2, 3, 5\} \). Now, find the intersection with \( C \): \( (A \cup B) \cap C = \{1, 2, 3, 5\} \cap \{2, 3, 4, 5\} = \{2, 3, 5\} \Rightarrow n((A \cup B) \cap C) = 3 \). \[ P(A \cup B | C) = \frac{n((A \cup B) \cap C)}{n(C)} = \frac{3}{4} \]
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