The probabilities that the events A and B occur are 0.3 and 0.4 respectively. The probability that both A and B occur simultaneously is 0.15. What is the probability that neither A nor B occurs?

Answer

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

### Step 1: Identify the given probabilities
- Probability of event A occurring, \( P(A) = 0.3 \)
- Probability of event B occurring, \( P(B) = 0.4 \)
- Probability of both events A and B occurring simultaneously, \( P(A \cap B) = 0.15 \)

### Step 2: Use the formula for the probability of the union of two events
The probability that at least one of the events A or B occurs is given by the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ P(A \cup B) = 0.3 + 0.4 - 0.15 \]

### Step 4: Calculate the probability of the union
Calculating the right-hand side:
\[ P(A \cup B) = 0.7 - 0.15 = 0.55 \]

### Step 5: Find the probability that neither A nor B occurs
The probability that neither A nor B occurs is the complement of the probability that at least one of them occurs:
\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(\text{neither A nor B}) = 1 - 0.55 = 0.45 \]

### Final Answer
The probability that neither A nor B occurs is \( \boxed{0.45} \).

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