The probability student A passing an examination is `(3)/(5)`, student B passing an examination is `(4)/(5)`. Find the probability only one of them will pass the examination, assuming the events 'A passses and 'B passes' are independent.

Answer

To solve the problem of finding the probability that only one of the two students, A or B, passes the examination, we can follow these steps:

### Step 1: Identify the probabilities of passing and failing for each student.
- The probability of student A passing the examination, \( P(A) = \frac{3}{5} \).
- The probability of student B passing the examination, \( P(B) = \frac{4}{5} \).

### Step 2: Calculate the probabilities of failing for each student.
- The probability of student A failing the examination, \( P(A') = 1 - P(A) = 1 - \frac{3}{5} = \frac{2}{5} \).
- The probability of student B failing the examination, \( P(B') = 1 - P(B) = 1 - \frac{4}{5} = \frac{1}{5} \).

### Step 3: Use the formula for the probability of only one passing.
Since the events are independent, the probability that only one of them passes can be calculated using the following two scenarios:
1. Student A passes and student B fails.
2. Student A fails and student B passes.

The formula is:
\[ P(\text{only A passes}) + P(\text{only B passes}) = P(A) \cdot P(B') + P(A') \cdot P(B) \]

### Step 4: Substitute the values into the formula.
- For the first scenario (A passes, B fails):
\[ P(A) \cdot P(B') = \frac{3}{5} \cdot \frac{1}{5} = \frac{3}{25} \]
- For the second scenario (A fails, B passes):
\[ P(A') \cdot P(B) = \frac{2}{5} \cdot \frac{4}{5} = \frac{8}{25} \]

### Step 5: Add the probabilities from both scenarios.
\[ P(\text{only one passes}) = \frac{3}{25} + \frac{8}{25} = \frac{11}{25} \]

### Final Answer:
Thus, the probability that only one of them will pass the examination is \( \frac{11}{25} \).

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