NIMCET 2026 Mathematics Question 21 with Solution

Question 21

Mathematics Probability Hard

If \( 0 < P(A) < 1 \), \( 0 < P(B) < 1 \), and \( P(A \cup B) = P(A) + P(B) - P(A)P(B) \), then

(A) \( P(B \mid A) = P(B) - P(A) \)

(B) \( P(A' - B') = P(A') - P(B') \)

(C) \( P(A \cup B)' = P(A)' P(B)' \)

(D) \( P(A \mid B) = P(A) - P(B) \)

View Dynamic Solution & Explanation ▼

Correct Solution: Option C

Step-by-step Solution:

Here is the correctly formatted text:
Since, \( P(A \cap B) = P(A) \cdot P(B) \)
It means \( A \) and \( B \) are independent events, so \( A' \) and \( B' \) are also independent. \[ \therefore P(A \cup B)' = P(A' \cap B') = P(A)' \cdot P(B)' \] Alternative Solution: \[ P(A \cup B)' = 1 - P(A \cup B) = 1 - \{ P(A) + P(B) - P(A) \cdot P(B) \} \] \[ = \{ 1 - P(A) \} \cdot \{ 1 - P(B) \} = P(A)' \cdot P(B)' \]