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 does not achieve success, the chance she studied for 4 hours is?

Options

A. `(15)/(26)`
B. `(17)/(26)`
C. `(19)/(26)`
D. `(21)/(26)`

Answer

To solve the problem step by step, we will use the concepts of conditional probability and Bayes' theorem.

### 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.
- \( B \): Event that she does not achieve success.

From the problem, we know:
- \( P(A_1) = 0.1 \)
- \( P(A_2) = 0.2 \)
- \( P(A_3) = 0.7 \)

### Step 2: Calculate the probabilities of not achieving success
The probabilities of not achieving success given the hours studied are:
- \( P(B | A_1) = 1 - 0.8 = 0.2 \) (for 10 hours)
- \( P(B | A_2) = 1 - 0.6 = 0.4 \) (for 7 hours)
- \( P(B | A_3) = 1 - 0.4 = 0.6 \) (for 4 hours)

### Step 3: Calculate the total probability of not achieving success
Using the law of total probability:
\[ P(B) = P(B | A_1)P(A_1) + P(B | A_2)P(A_2) + P(B | A_3)P(A_3) \]
Substituting the values:
\[ P(B) = (0.2)(0.1) + (0.4)(0.2) + (0.6)(0.7) \]
Calculating each term:
- \( (0.2)(0.1) = 0.02 \)
- \( (0.4)(0.2) = 0.08 \)
- \( (0.6)(0.7) = 0.42 \)

Adding these:
\[ P(B) = 0.02 + 0.08 + 0.42 = 0.52 \]

### Step 4: Calculate the conditional probability of studying for 4 hours given that she did not achieve success
We want to find \( P(A_3 | B) \). By Bayes' theorem:
\[ P(A_3 | B) = \frac{P(B | A_3)P(A_3)}{P(B)} \]
Substituting the values:
\[ P(A_3 | B) = \frac{P(B | A_3)P(A_3)}{P(B)} = \frac{(0.6)(0.7)}{0.52} \]
Calculating the numerator:
\[ (0.6)(0.7) = 0.42 \]
Thus:
\[ P(A_3 | B) = \frac{0.42}{0.52} \]

### Step 5: Simplify the fraction
To simplify \( \frac{0.42}{0.52} \):
\[ P(A_3 | B) = \frac{42}{52} = \frac{21}{26} \]

### Final Answer
The probability that she studied for 4 hours given that she did not achieve success is \( \frac{21}{26} \).

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