In a multiple choice test, an examinee either knows the correct answer with probability p. or guesses with probability 1-p. The probability of answering a question correctly is `(1)/(m)`. If he or she merely guesses.IF the examinee answers a question correctly. the probability that he or she really knows the answer is :

Options

A. `(mp)/(1+mp)`
B. `(mp)/(1+ (m-1)p)`
C. `((m-1)p)/(1+(m-1)p)`
D. `((m-1)p)/(1+mp)`

Answer

To solve the problem, we need to find the probability that the examinee really knows the answer given that they answered correctly. This is a classic application of Bayes' Theorem.

### Step-by-Step Solution:

1. **Define the Events**:
- Let \( K \) be the event that the examinee knows the answer.
- Let \( C \) be the event that the examinee answers correctly.

2. **Given Probabilities**:
- The probability that the examinee knows the answer: \( P(K) = p \).
- The probability that the examinee guesses: \( P(K^c) = 1 - p \).
- The probability of answering correctly if they guess: \( P(C | K^c) = \frac{1}{m} \).
- If the examinee knows the answer, they will definitely answer correctly: \( P(C | K) = 1 \).

3. **Total Probability of Answering Correctly**:
- We can find \( P(C) \) using the law of total probability:
\[ P(C) = P(C | K) \cdot P(K) + P(C | K^c) \cdot P(K^c) \]
Substituting the known values:
\[ P(C) = 1 \cdot p + \frac{1}{m} \cdot (1 - p) = p + \frac{1 - p}{m} \]

4. **Using Bayes' Theorem**:
- We want to find \( P(K | C) \), the probability that the examinee knows the answer given that they answered correctly:
\[ P(K | C) = \frac{P(C | K) \cdot P(K)}{P(C)} \]
Substituting the values we have:
\[ P(K | C) = \frac{1 \cdot p}{p + \frac{1 - p}{m}} \]

5. **Simplifying the Expression**:
- To simplify the expression, we can multiply the numerator and denominator by \( m \):
\[ P(K | C) = \frac{m \cdot p}{m \cdot p + 1 - p} \]
- This gives us the final result:
\[ P(K | C) = \frac{m \cdot p}{(m - 1) \cdot p + 1} \]

### Final Answer:
The probability that the examinee really knows the answer given that they answered correctly is:
\[ P(K | C) = \frac{m \cdot p}{(m - 1) \cdot p + 1} \]