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

Step 1: Understanding the Question
The question asks for the probability of the union of two events, \(A\) and \(B\), given their individual probabilities and the fact that they are independent.
Step 2: Key Formula or Approach
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] For independent events, the probability of their intersection is the product of their individual probabilities:
\[ P(A \cap B) = P(A) \times P(B) \] Step 3: Detailed Explanation
First, we calculate the probability of the intersection \(P(A \cap B)\) using the independence property.
Given \( P(A) = 0.4 \) and \( P(B) = 0.5 \):
\[ P(A \cap B) = 0.4 \times 0.5 = 0.20 \] Next, we substitute this value into the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] \[ P(A \cup B) = 0.4 + 0.5 - 0.20 \] \[ P(A \cup B) = 0.9 - 0.20 = 0.70 \] Step 4: Final Answer
The value of \( P(A \cup B) \) is \(0.70\).