Question

The chance of India winning a cricket match against Australia is \( \tfrac{1}{6} \). What is the minimum number of matches India should play against Australia so that there is a fair chance of winning at least one match?

• A. 3
• B. 4
• C. 5
• D. 6
• E. None of the above

Hint:

In “at least one” probability problems, always use the complement rule: P(at least one success) = 1 − P(no success). This avoids long casewise calculations.

Answer 1

The Correct Option is B

Step 1: Define probability of winning and losing.
Probability of India winning one match = \( \tfrac{1}{6} \).
Probability of India losing one match = \( 1 - \tfrac{1}{6} = \tfrac{5}{6} \).
Step 2: Probability of losing all matches.
If India plays \(x\) matches, the probability that it loses all \(x\) matches is \[ \left(\tfrac{5}{6}\right)^x. \] Step 3: Probability of winning at least one match.
Probability of winning at least one = \(1 - \left(\tfrac{5}{6}\right)^x\).
Step 4: Fair chance condition.
We want this probability to be at least one-half: \[ 1 - \left(\tfrac{5}{6}\right)^x \geq \tfrac{1}{2}. \] This simplifies to: \[ \left(\tfrac{5}{6}\right)^x \leq \tfrac{1}{2}. \] Step 5: Test values of \(x\).
- For \(x = 3\):
\(\left(\tfrac{5}{6}\right)^3 = \tfrac{125}{216} \approx 0.579>0.5\).
So probability of winning at least one = 1 − 0.579 = 0.421<0.5. Not sufficient.
- For \(x = 4\):
\(\left(\tfrac{5}{6}\right)^4 = \tfrac{625}{1296} \approx 0.482<0.5\).
So probability of winning at least one = 1 − 0.482 = 0.518>0.5. This satisfies the condition.
Step 6: Conclude.
Thus, India must play at least 4 matches to have a fair chance of winning at least one match. \[ \boxed{4 \; \text{(Option B)}} \]