The probability of an event A occuring is 0.5 and of B is 0.3 If A and B are mutually exclusive events , then find the probability of neither A nor B occurring.

Answer

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

### Step 1: Understand the probabilities of events A and B
We are given:
- Probability of event A occurring, P(A) = 0.5
- Probability of event B occurring, P(B) = 0.3

### Step 2: Use the property of mutually exclusive events
Since events A and B are mutually exclusive, this means that they cannot occur at the same time. Therefore, the probability of both A and B occurring together, P(A ∩ B), is 0.

### Step 3: Calculate the probability of either A or B occurring
To find the probability of either A or B occurring, we can use the formula for mutually exclusive events:
\[ P(A \cup B) = P(A) + P(B) \]

Substituting the values:
\[ P(A \cup B) = 0.5 + 0.3 = 0.8 \]

### Step 4: Find the probability of neither A nor B occurring
The probability of neither A nor B occurring is the complement of the probability of either A or B occurring. This can be calculated as:
\[ P(A' \cap B') = 1 - P(A \cup B) \]

Substituting the value we found:
\[ P(A' \cap B') = 1 - 0.8 = 0.2 \]

### Final Answer
The probability of neither A nor B occurring is:
\[ P(A' \cap B') = 0.2 \]

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