Question

If \( P(A) = 0.4 \), \( P(B) = 0.5 \), and \(A\) and \(B\) are independent events, what is the value of \( P(A \cup B) \)?

• A. \(0.60\)
• B. \(0.65\)
• C. \(0.70\)
• D. \(0.75\)

Hint:

For independent events, remember that the probability of both events occurring together is the product of their probabilities: \[ P(A \cap B) = P(A)P(B) \] This simplifies many probability calculations.

Answer 1

The Correct Option is C

Concept:
For any two events \(A\) and \(B\), the probability of their union is given by: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] If \(A\) and \(B\) are independent events, then: \[ P(A \cap B) = P(A)P(B) \] Thus, the formula becomes: \[ P(A \cup B) = P(A) + P(B) - P(A)P(B) \]
Step 1: Find \(P(A \cap B)\) using independence.
\[ P(A \cap B) = P(A)P(B) \] Substitute the given values: \[ P(A \cap B) = 0.4 \times 0.5 = 0.20 \]
Step 2: Apply the union formula.
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substitute the values: \[ P(A \cup B) = 0.4 + 0.5 - 0.20 \] \[ P(A \cup B) = 0.70 \] \[ \therefore P(A \cup B) = 0.70 \]