The probability that India wins a cricket test match against England is 1/3. if india and england play three test matches, the probability that India will win at least one test match is_____

Answer

To solve the problem, we need to find the probability that India wins at least one test match out of three matches against England, given that the probability of India winning a single match is \( \frac{1}{3} \).

### Step-by-Step Solution:

1. **Identify the Probability of Winning and Losing:**
- The probability that India wins a match, \( P(W) = \frac{1}{3} \).
- Therefore, the probability that India does not win a match (loses or draws), \( P(L) = 1 - P(W) = 1 - \frac{1}{3} = \frac{2}{3} \).

2. **Calculate the Probability of Losing All Matches:**
- If India plays 3 matches, the probability that India loses all three matches is calculated as:
\[ P(\text{Losing all 3 matches}) = P(L) \times P(L) \times P(L) = \left(\frac{2}{3}\right)^3 \]
- Calculate \( \left(\frac{2}{3}\right)^3 \):
\[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]

3. **Calculate the Probability of Winning at Least One Match:**
- The probability that India wins at least one match is the complement of the probability that India loses all matches:
\[ P(\text{Winning at least 1 match}) = 1 - P(\text{Losing all 3 matches}) = 1 - \frac{8}{27} \]
- Simplifying this gives:
\[ P(\text{Winning at least 1 match}) = 1 - \frac{8}{27} = \frac{27 - 8}{27} = \frac{19}{27} \]

### Final Answer:
The probability that India will win at least one test match is \( \frac{19}{27} \).

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