Question

The probability of occurrence of two events \(A\) and \(B\) are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. The probability that neither \(A\) occurs nor \(B\) occurs is:

• A. 0.61
• B. 0.11
• C. 0.39
• D. 0.89

Hint:

Always calculate the union \(P(A \cup B)\) first, and then subtract it from 1 to find the probability of neither event occurring.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
To find the probability that neither of two events occurs, we find the complement of the probability that at least one of the events occurs.

Step 2: Key Formula or Approach:
The probability that at least one event occurs is given by the addition theorem:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
The probability that neither event occurs is:
\[ P(\text{neither}) = P(A' \cap B') = 1 - P(A \cup B) \]

Step 3: Detailed Explanation:
We are given:
- Probability of event \(A\), \(P(A) = 0.25\)
- Probability of event \(B\), \(P(B) = 0.50\)
- Probability of their joint occurrence, \(P(A \cap B) = 0.14\)
First, calculate the probability of the union, \(P(A \cup B)\):
\[ P(A \cup B) = 0.25 + 0.50 - 0.14 \]
\[ P(A \cup B) = 0.75 - 0.14 = 0.61 \]
Now, calculate the probability that neither event occurs:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
\[ P(A' \cap B') = 1 - 0.61 = 0.39 \]
The probability that neither event occurs is 0.39.
This matches the third option.

Step 4: Final Answer:
Therefore, the correct option is (C).