If A and B are two independent events such that `P(A) = 0.40 , P(B) = 0.50`.Find P(neither A nor B).

Options

A. `0.90`
B. `0.10`
C. `0.2`
D. `0.3`

Answer

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

### Step 1: Understand the Problem
We need to find the probability of neither A nor B occurring. This can be expressed in terms of the complements of A and B.

### Step 2: Use the Complement Rule
The probability of neither A nor B can be expressed as:
\[ P(\text{neither A nor B}) = P(A') \cap P(B') \]
Where \( A' \) is the complement of A and \( B' \) is the complement of B.

### Step 3: Calculate the Complements
We know:
- \( P(A) = 0.40 \)
- \( P(B) = 0.50 \)

Thus, the complements are:
\[ P(A') = 1 - P(A) = 1 - 0.40 = 0.60 \]
\[ P(B') = 1 - P(B) = 1 - 0.50 = 0.50 \]

### Step 4: Use the Independence of Events
Since A and B are independent events, the probability of both A' and B' occurring is:
\[ P(A' \cap B') = P(A') \times P(B') \]

### Step 5: Calculate the Probability
Now we can substitute the values:
\[ P(A' \cap B') = P(A') \times P(B') = 0.60 \times 0.50 \]

Calculating this gives:
\[ P(A' \cap B') = 0.30 \]

### Final Answer
Thus, the probability of neither A nor B occurring is:
\[ P(\text{neither A nor B}) = 0.30 \]

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