Two events A and B have probability 0.28 and 0.55 respectively. The probability that A and B occur simultaneously is 0.14. Find the probability that neither A nor B occurs

Options

A. `0.39`
B. `0.41`
C. `0.4`
D. `0.31`

Answer

To find the probability that neither event A nor event B occurs, we can follow these steps:

### Step 1: Understand the given probabilities
We are given:
- Probability of event A, \( P(A) = 0.28 \)
- Probability of event B, \( P(B) = 0.55 \)
- Probability that both A and B occur simultaneously, \( P(A \cap B) = 0.14 \)

### Step 2: Use the formula for the union of two events
To find the probability that either A or B occurs (the union of A and B), we can use the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Substitute the values into the formula
Now, substituting the known values into the formula:
\[ P(A \cup B) = 0.28 + 0.55 - 0.14 \]

### Step 4: Calculate the probability of A union B
Now, perform the calculation:
\[ P(A \cup B) = 0.28 + 0.55 - 0.14 = 0.69 \]

### Step 5: Find the probability that neither A nor B occurs
The probability that neither A nor B occurs is given by:
\[ P(A' \cap B') = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(A' \cap B') = 1 - 0.69 = 0.31 \]

### Final Answer
Thus, the probability that neither A nor B occurs is:
\[ \boxed{0.31} \]

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