India and Pakistan play a 5 match test series of hockey, the probability that India wins at least three matches is

Options

A. `1/2`
B. `3/5`
C. `4/5`
D. None of these

Answer

To find the probability that India wins at least three matches in a 5-match test series against Pakistan, we can use the binomial probability formula.

### Step-by-Step Solution:

1. **Define the Probability of Winning and Losing:**
Let \( P \) be the probability that India wins a match. Since there are only two outcomes (win or lose), if we assume the probability of winning a match is \( \frac{1}{2} \), then the probability of losing a match \( Q \) is also \( \frac{1}{2} \).

2. **Identify the Required Outcomes:**
We need to calculate the probability of India winning at least 3 matches. This includes the scenarios where India wins 3, 4, or 5 matches.

3. **Use the Binomial Probability Formula:**
The binomial probability formula is given by:
\[ P(X = k) = \binom{n}{k} P^k Q^{n-k} \]
where:
- \( n \) = total number of matches (5)
- \( k \) = number of matches won
- \( P \) = probability of winning a match
- \( Q \) = probability of losing a match

4. **Calculate the Probability for Winning 3 Matches:**
\[ P(X = 3) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{5-3} = \binom{5}{3} \left(\frac{1}{2}\right)^5 \]
\[ = 10 \cdot \frac{1}{32} = \frac{10}{32} \]

5. **Calculate the Probability for Winning 4 Matches:**
\[ P(X = 4) = \binom{5}{4} \left(\frac{1}{2}\right)^4 \left(\frac{1}{2}\right)^{5-4} = \binom{5}{4} \left(\frac{1}{2}\right)^5 \]
\[ = 5 \cdot \frac{1}{32} = \frac{5}{32} \]

6. **Calculate the Probability for Winning 5 Matches:**
\[ P(X = 5) = \binom{5}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{5-5} = \binom{5}{5} \left(\frac{1}{2}\right)^5 \]
\[ = 1 \cdot \frac{1}{32} = \frac{1}{32} \]

7. **Sum the Probabilities:**
Now, we sum the probabilities of winning 3, 4, and 5 matches:
\[ P(X \geq 3) = P(X = 3) + P(X = 4) + P(X = 5) \]
\[ = \frac{10}{32} + \frac{5}{32} + \frac{1}{32} = \frac{16}{32} = \frac{1}{2} \]

8. **Final Result:**
The probability that India wins at least 3 matches is \( \frac{1}{2} \).