A JEE aspirant estimates that she will be successful with an 80 percent chance if she studies 10 hours per day, with a 60 percent chance if she studies 7 hours per day and with 40 percent chance if she studies 4 hours per day. She further believes that she will study 10 hours, 7 hours and 4 hours per day with probabilities 0.1, 0.2 and 0.7 respectively. Given that she will achieve success, the chance she studied for 4 hours is?

Options

A. `(1)/(12)`
B. `(5)/(12)`
C. `(7)/(12)`
D. `(11)/(12)`

Answer

To solve the problem step by step, we will use Bayes' theorem. We need to find the probability that the aspirant studied for 4 hours given that she achieved success.

### Step 1: Define the events
Let:
- \( A_1 \): Event that she studies for 10 hours.
- \( A_2 \): Event that she studies for 7 hours.
- \( A_3 \): Event that she studies for 4 hours.
- \( S \): Event that she is successful.

### Step 2: Given probabilities
From the problem, we have:
- \( P(A_1) = 0.1 \), \( P(S|A_1) = 0.8 \)
- \( P(A_2) = 0.2 \), \( P(S|A_2) = 0.6 \)
- \( P(A_3) = 0.7 \), \( P(S|A_3) = 0.4 \)

### Step 3: Calculate the total probability of success, \( P(S) \)
Using the law of total probability:
\[ P(S) = P(S|A_1)P(A_1) + P(S|A_2)P(A_2) + P(S|A_3)P(A_3) \]

Substituting the values:
\[ P(S) = (0.8 \times 0.1) + (0.6 \times 0.2) + (0.4 \times 0.7) \]
\[ = 0.08 + 0.12 + 0.28 = 0.48 \]

### Step 4: Use Bayes' theorem to find \( P(A_3|S) \)
We need to find \( P(A_3|S) \):
\[ P(A_3|S) = \frac{P(S|A_3)P(A_3)}{P(S)} \]

Substituting the known values:
\[ P(A_3|S) = \frac{P(S|A_3)P(A_3)}{P(S)} = \frac{0.4 \times 0.7}{0.48} \]
\[ = \frac{0.28}{0.48} \]

### Step 5: Simplify the fraction
\[ P(A_3|S) = \frac{28}{48} = \frac{7}{12} \]

### Final Answer
Thus, the probability that she studied for 4 hours given that she achieved success is \( \frac{7}{12} \).

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