Shubham has `75%` chance of attending the annual meet. Shikha has a `90%` chance if Shubham also attends otherwise she has a `40%` chance of attending the meet. If I go to the annual meet and see Shikha there, then the probability the Shubam is also there, is

Options

A. `(27)/(31)`
B. `(19)/(30)`
C. `(1)/(5)`
D. `(9)/(10)`

Answer

To solve the problem, we need to find the probability that Shubham is at the annual meet given that Shikha is there. We will use Bayes' theorem to find this probability.

### Step 1: Define Events
Let:
- \( A \): Shubham attends the annual meet.
- \( B \): Shikha attends the annual meet.

### Step 2: Given Probabilities
From the problem, we have:
- \( P(A) = 0.75 \) (Shubham's probability of attending)
- \( P(B|A) = 0.90 \) (Shikha's probability of attending given Shubham attends)
- \( P(B|A^c) = 0.40 \) (Shikha's probability of attending given Shubham does not attend)

### Step 3: Find \( P(A^c) \)
The probability that Shubham does not attend is:
\[ P(A^c) = 1 - P(A) = 1 - 0.75 = 0.25 \]

### Step 4: Find \( P(B) \)
We can find the total probability that Shikha attends the meet using the law of total probability:
\[ P(B) = P(B|A)P(A) + P(B|A^c)P(A^c) \]
Substituting the known values:
\[ P(B) = (0.90)(0.75) + (0.40)(0.25) \]
Calculating each term:
\[ P(B) = 0.675 + 0.10 = 0.775 \]

### Step 5: Apply Bayes' Theorem
We want to find \( P(A|B) \), the probability that Shubham is at the meet given that Shikha is there:
\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]
Substituting the values we have:
\[ P(A|B) = \frac{(0.90)(0.75)}{0.775} \]
Calculating the numerator:
\[ P(A|B) = \frac{0.675}{0.775} \]

### Step 6: Simplify the Fraction
Now we simplify:
\[ P(A|B) = \frac{675}{775} = \frac{27}{31} \]

### Final Answer
Thus, the probability that Shubham is also at the annual meet given that Shikha is there is:
\[ \boxed{\frac{27}{31}} \]