Question

If P(A) = 0.59, P(B) = 0.30 and P(A ∩ B) = 0.21 then P( A' ∩ B') =

• A. 0.11
• B. 0.38
• C. 0.32
• D. 0.35

Answer 1

The Correct Option is C

We are given the following probabilities: \[ P(A) = 0.59, \quad P(B) = 0.30, \quad P(A \cap B) = 0.21 \] The probability of the complement \( P(A' \cap B') \), i.e., the probability that neither A nor B occurs, is given by the formula: \[ P(A' \cap B') = 1 - P(A \cup B) \] Using the formula for the union of two sets: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting the known values: \[ P(A \cup B) = 0.59 + 0.30 - 0.21 = 0.68 \] Therefore: \[ P(A' \cap B') = 1 - 0.68 = 0.32 \] 

Answer 2

We are given that P(A) = 0.59, P(B) = 0.30, and P(A ∩ B) = 0.21.

We need to find P(A' ∩ B').

By De Morgan's Law, we know that \(A' ∩ B' = (A ∪ B)'\). Therefore, \(P(A' ∩ B') = P((A ∪ B)')\).

We also know that \(P((A ∪ B)') = 1 - P(A ∪ B)\).

The formula for \(P(A ∪ B)\) is \(P(A ∪ B) = P(A) + P(B) - P(A ∩ B)\).

Substituting the given values, we get \(P(A ∪ B) = 0.59 + 0.30 - 0.21 = 0.89 - 0.21 = 0.68\).

Now, we can find \(P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.68 = 0.32\).

Therefore, P(A' ∩ B') = 0.32.