India plays, 2,3 and 5 matches against Pakistan, Srilanka and West Indies respectively. Probability that India win a match against Pakistan, Srilanka and West Indies is 0.6,0.5 and 0.4 respecitvely. If india win a match , what is the probability that it was against Pakistan.

Answer

To solve the problem, we will use Bayes' theorem to find the probability that India played against Pakistan given that they won a match.

### Step-by-step Solution:

1. **Identify the Matches Played:**
- India plays 2 matches against Pakistan.
- India plays 3 matches against Sri Lanka.
- India plays 5 matches against West Indies.
- Total matches = 2 + 3 + 5 = 10 matches.

2. **Calculate the Probability of Playing Against Each Team:**
- Probability of playing against Pakistan, \( P(P) = \frac{2}{10} = 0.2 \)
- Probability of playing against Sri Lanka, \( P(SL) = \frac{3}{10} = 0.3 \)
- Probability of playing against West Indies, \( P(WI) = \frac{5}{10} = 0.5 \)

3. **Calculate the Probability of Winning Against Each Team:**
- Probability of winning against Pakistan, \( P(W|P) = 0.6 \)
- Probability of winning against Sri Lanka, \( P(W|SL) = 0.5 \)
- Probability of winning against West Indies, \( P(W|WI) = 0.4 \)

4. **Calculate the Total Probability of Winning, \( P(W) \):**
Using the law of total probability:
\[ P(W) = P(W|P) \cdot P(P) + P(W|SL) \cdot P(SL) + P(W|WI) \cdot P(WI) \]
Substituting the values:
\[ P(W) = (0.6 \cdot 0.2) + (0.5 \cdot 0.3) + (0.4 \cdot 0.5) \]
\[ P(W) = 0.12 + 0.15 + 0.20 = 0.47 \]

5. **Apply Bayes' Theorem:**
We want to find \( P(P|W) \), the probability that India played against Pakistan given that they won:
\[ P(P|W) = \frac{P(W|P) \cdot P(P)}{P(W)} \]
Substituting the values:
\[ P(P|W) = \frac{0.6 \cdot 0.2}{0.47} \]
\[ P(P|W) = \frac{0.12}{0.47} \]

6. **Calculate the Final Probability:**
\[ P(P|W) \approx 0.2553 \]

### Final Answer:
The probability that India played against Pakistan given that they won a match is approximately \( 0.2553 \) or \( \frac{12}{47} \).