The probability of a candidate passing an examination at any one attempt is `3/5`.He carries on entering until he passes and each entry costs him Rs. 1. The expected cost of his passing the exam is (in Rs)

Options

A. `25/9`
B. `5/3`
C. `125/27`
D. `625/81`

Answer

To find the expected cost of a candidate passing the examination, we can use the concept of expected value in probability.

### Step-by-Step Solution:

1. **Identify the Probability of Passing and Failing:**
- The probability of passing the exam in one attempt, \( P \), is given as \( \frac{3}{5} \).
- The probability of failing the exam, \( F \), is \( 1 - P = 1 - \frac{3}{5} = \frac{2}{5} \).

2. **Define the Expected Number of Attempts:**
- The candidate continues to attempt the exam until he passes. The number of attempts until the first success (passing) follows a geometric distribution.
- The expected number of attempts \( E(X) \) to get the first success in a geometric distribution is given by the formula:
\[ E(X) = \frac{1}{P} \]

3. **Calculate the Expected Number of Attempts:**
- Substituting the value of \( P \):
\[ E(X) = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]

4. **Calculate the Expected Cost:**
- Since each attempt costs Rs. 1, the expected cost \( E(C) \) of passing the exam is equal to the expected number of attempts:
\[ E(C) = E(X) \times \text{Cost per attempt} = \frac{5}{3} \times 1 = \frac{5}{3} \text{ Rs} \]

5. **Final Result:**
- The expected cost of passing the exam is \( \frac{5}{3} \) Rs, which can also be expressed as approximately Rs. 1.67.

### Summary:
The expected cost of passing the exam is \( \frac{5}{3} \) Rs.

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