The probability of A to fail in an examination is `1/5` and that of B is `3/10`. If the probability that both of them fails is `3/50`, then the probability that only one of A and B fails, is

Options

A. `1/2`
B. `11/25`
C. `19/50`
D. `12/25`

Answer

To solve the problem, we need to find the probability that only one of A or B fails in the examination.

Let's denote:
- \( P(A) \): Probability that A fails = \( \frac{1}{5} \)
- \( P(B) \): Probability that B fails = \( \frac{3}{10} \)
- \( P(A \cap B) \): Probability that both A and B fail = \( \frac{3}{50} \)

### Step 1: Calculate the probabilities of passing
First, we need to find the probabilities that A and B pass:
- \( P(A') = 1 - P(A) = 1 - \frac{1}{5} = \frac{4}{5} \)
- \( P(B') = 1 - P(B) = 1 - \frac{3}{10} = \frac{7}{10} \)

### Step 2: Check for independence
To check if A and B are independent events, we need to verify if:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{1}{5} \cdot \frac{3}{10} = \frac{3}{50} \]
Since \( P(A \cap B) = \frac{3}{50} \) is equal to \( P(A) \cdot P(B) \), A and B are independent.

### Step 3: Calculate the probability that only one of them fails
The probability that only one of A or B fails can be calculated using the formula:
\[ P(\text{only A fails}) + P(\text{only B fails}) \]
This can be expressed as:
\[ P(A \cap B') + P(A' \cap B) \]
Using the independence of A and B:
- \( P(A \cap B') = P(A) \cdot P(B') = \frac{1}{5} \cdot \frac{7}{10} = \frac{7}{50} \)
- \( P(A' \cap B) = P(A') \cdot P(B) = \frac{4}{5} \cdot \frac{3}{10} = \frac{12}{50} \)

### Step 4: Combine the probabilities
Now, we add these two probabilities:
\[ P(\text{only A fails}) + P(\text{only B fails}) = \frac{7}{50} + \frac{12}{50} = \frac{19}{50} \]

### Final Answer
Thus, the probability that only one of A and B fails is:
\[ \frac{19}{50} \]

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