If P (A) =0.4,P(B)=0.8 and P(`A cap B)=0.3` then the probability that exactly one of them occurs is

Options

A. 0.2
B. 0.4
C. 0.6
D. 0.1

Answer

To find the probability that exactly one of the events A or B occurs, we can follow these steps:

### Step 1: Understand the Problem
We are given:
- \( P(A) = 0.4 \)
- \( P(B) = 0.8 \)
- \( P(A \cap B) = 0.3 \)

We need to find the probability that exactly one of the events A or B occurs.

### Step 2: Define the Events
The event that exactly one of A or B occurs can be expressed as:
- Only A occurs: \( A \) occurs and \( B \) does not occur.
- Only B occurs: \( B \) occurs and \( A \) does not occur.

### Step 3: Calculate the Probability of Only A Occurring
To find the probability that only A occurs, we subtract the probability of both A and B occurring from the probability of A:
\[ P(\text{Only A}) = P(A) - P(A \cap B) = 0.4 - 0.3 = 0.1 \]

### Step 4: Calculate the Probability of Only B Occurring
To find the probability that only B occurs, we subtract the probability of both A and B occurring from the probability of B:
\[ P(\text{Only B}) = P(B) - P(A \cap B) = 0.8 - 0.3 = 0.5 \]

### Step 5: Combine the Probabilities
The probability that exactly one of A or B occurs is the sum of the probabilities of only A occurring and only B occurring:
\[ P(\text{Exactly one of A or B}) = P(\text{Only A}) + P(\text{Only B}) = 0.1 + 0.5 = 0.6 \]

### Final Answer
Thus, the probability that exactly one of them occurs is \( \boxed{0.6} \).

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