In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is p. The probability that his answer is correct, given that he copied it, is 1/8. If the probability that he knew the answer to question, given that he correctly answered is 8/11, then value of p is

Options

A. `1/2`
B. `1/3`
C. `1/6`
D. `1/12`

Answer

To solve the problem step by step, we will use the information provided and apply the concepts of probability.

### Step 1: Define the probabilities
Let:
- \( P(G) = \frac{1}{3} \) (Probability of guessing)
- \( P(C) = p \) (Probability of copying)
- \( P(K) = 1 - P(G) - P(C) = 1 - \frac{1}{3} - p = \frac{2}{3} - p \) (Probability of knowing the answer)

### Step 2: Use the given probabilities
We know:
- \( P(A_C | C) = \frac{1}{8} \) (Probability of a correct answer given that he copied)
- \( P(K | A_C) = \frac{8}{11} \) (Probability of knowing the answer given that he answered correctly)

### Step 3: Apply Bayes' Theorem
Using Bayes' theorem, we can express \( P(K | A_C) \) as follows:

\[ P(K | A_C) = \frac{P(A_C | K) \cdot P(K)}{P(A_C)} \]

Where \( P(A_C) \) can be expressed as:

\[ P(A_C) = P(A_C | G) \cdot P(G) + P(A_C | C) \cdot P(C) + P(A_C | K) \cdot P(K) \]

### Step 4: Calculate \( P(A_C | G) \)
If he guesses, the probability of getting the answer correct is:

\[ P(A_C | G) = \frac{1}{4} \]

### Step 5: Substitute known values into \( P(A_C) \)
Now substituting the known values into the equation for \( P(A_C) \):

\[ P(A_C) = P(A_C | G) \cdot P(G) + P(A_C | C) \cdot P(C) + P(A_C | K) \cdot P(K) \]

Substituting the values we have:

\[ P(A_C) = \left(\frac{1}{4} \cdot \frac{1}{3}\right) + \left(\frac{1}{8} \cdot p\right) + \left(1 \cdot \left(\frac{2}{3} - p\right)\right) \]

Calculating this gives:

\[ P(A_C) = \frac{1}{12} + \frac{p}{8} + \left(\frac{2}{3} - p\right) \]

### Step 6: Set up the equation using Bayes' Theorem
Now we can set up the equation using the values we have:

\[ \frac{8}{11} = \frac{P(A_C | K) \cdot P(K)}{P(A_C)} \]

Since \( P(A_C | K) = 1 \):

\[ \frac{8}{11} = \frac{1 \cdot \left(\frac{2}{3} - p\right)}{P(A_C)} \]

### Step 7: Substitute \( P(A_C) \) into the equation
Substituting \( P(A_C) \):

\[ \frac{8}{11} = \frac{\frac{2}{3} - p}{\frac{1}{12} + \frac{p}{8} + \frac{2}{3} - p} \]

### Step 8: Solve for \( p \)
Cross-multiplying gives:

\[ 8 \left(\frac{1}{12} + \frac{p}{8} + \frac{2}{3} - p\right) = 11 \left(\frac{2}{3} - p\right) \]

Expanding both sides:

\[ \frac{8}{12} + p + \frac{16}{3} - 8p = \frac{22}{3} - 11p \]

Combining like terms and solving for \( p \):

\[ \frac{2}{3} + p - 8p + 11p = \frac{22}{3} \]
\[ \frac{2}{3} + 4p = \frac{22}{3} \]
\[ 4p = \frac{22}{3} - \frac{2}{3} = \frac{20}{3} \]
\[ p = \frac{20}{12} = \frac{5}{3} \]

### Final Answer
Thus, the value of \( p \) is:

\[ p = \frac{1}{3} \]