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 step by step, we will use the concept of conditional probability and Bayes' theorem.

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

### Step 2: Given Probabilities
From the problem, we know:
- \( P(A) = 0.75 \) (Shubham has a 75% chance of attending)
- \( P(B|A) = 0.90 \) (If Shubham attends, Shikha has a 90% chance of attending)
- \( P(B|A') = 0.40 \) (If Shubham does not attend, Shikha has a 40% chance of attending)

Where \( A' \) is the complement of \( A \) (i.e., Shubham does not attend).

### Step 3: Calculate \( P(A') \)
Since \( P(A) + P(A') = 1 \):
\[ P(A') = 1 - P(A) = 1 - 0.75 = 0.25 \]

### Step 4: Use the Law of Total Probability to Find \( P(B) \)
We can find \( P(B) \) using the law of total probability:
\[ P(B) = P(B|A)P(A) + P(B|A')P(A') \]
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: Use Bayes' Theorem to Find \( P(A|B) \)
Now, we want to find \( P(A|B) \) (the probability that Shubham is there given that Shikha is there):
\[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \]
Substituting the known values:
\[ P(A|B) = \frac{(0.90)(0.75)}{0.775} \]
Calculating the numerator:
\[ P(A|B) = \frac{0.675}{0.775} \]
Calculating the final probability:
\[ P(A|B) \approx 0.87096 \]

### Final Result
Thus, the probability that Shubham is also there given that Shikha is there is approximately \( 0.871 \) or \( 87.1\% \).

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