ProbabilityPYQ Sept 25Question 4484 of 187
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Two persons are playing a set of matches. The winner of 4 matches is declared as the winner. Any player has 50% chance to win a match. The probability that the game comes to an end at the fourth match is

Options

A5/8
B4/8
C3/8
D1/8
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Correct Answer

✅ Option d — 1/8

All Options:

  • A5/8
  • B4/8
  • C3/8
  • D1/8

Detailed Solution & Explanation

Let the two players be X\displaystyle X and Y\displaystyle Y. The probability of either player winning any individual match is p=0.5=12\displaystyle p = 0.5 = \frac{1}{2}.
The game ends when one player wins 4 matches. For the game to end in exactly the 4th match, either:
1. **Player X\displaystyle X wins all the first 4 matches**:
The probability of X\displaystyle X winning the first, second, third, and fourth matches is:
P(X wins in 4)=(12)4=116P(X \text{ wins in 4}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16}
2. **Player Y\displaystyle Y wins all the first 4 matches**:
The probability of Y\displaystyle Y winning the first, second, third, and fourth matches is:
P(Y wins in 4)=(12)4=116P(Y \text{ wins in 4}) = \left(\frac{1}{2}\right)^4 = \frac{1}{16}
Since these two events are mutually exclusive (they cannot both win the series 4-0), the total probability that the game ends at the fourth match is the sum of their individual probabilities:
P(Ends at 4th match)=P(X wins in 4)+P(Y wins in 4)P(\text{Ends at 4th match}) = P(X \text{ wins in 4}) + P(Y \text{ wins in 4})
P(Ends at 4th match)=116+116=216=18P(\text{Ends at 4th match}) = \frac{1}{16} + \frac{1}{16} = \frac{2}{16} = \frac{1}{8}
Hence, **Option D** is the correct answer.

About This Chapter: Probability

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Probability Operations, Expected Value

A logic-heavy chapter dealing with random experiments, events (mutually exclusive, exhaustive), set theory probability, conditional probability, and Bayes' Theorem. It forms the basis for Theoretical Distributions.

View Official ICAI Syllabus

Exam Strategy Tip

Always draw a quick Venn Diagram or tree when faced with 'At least one' or 'Only A but not B' wording. It saves you from double-counting.

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