ProbabilityMTP June 2023 Series IQuestion 2829 of 295
All Questions

In a poker set there are 90\displaystyle 90 chips numbered from 1\displaystyle 1 to 90\displaystyle 90. Dan picks 3\displaystyle 3 chips random, one after the other, without replacement. What is the probability that the numbers on the chips, in the order that the picks them are in descending order?

Options

A13\displaystyle \frac{1}{3}
B130\displaystyle \frac{1}{30}
C16\displaystyle \frac{1}{6}
DNone
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Correct Answer

✅ Option c — 16\displaystyle \frac{1}{6}

All Options:

  • A13\displaystyle \frac{1}{3}
  • B130\displaystyle \frac{1}{30}
  • C16\displaystyle \frac{1}{6}
  • DNone

Detailed Solution & Explanation

Let the three chips picked at random without replacement have numbers X1\displaystyle X_1, X2\displaystyle X_2, and X3\displaystyle X_3. Since the chips are drawn without replacement, X1\displaystyle X_1, X2\displaystyle X_2, and X3\displaystyle X_3 are distinct numbers from the set {1,2,…,90}\displaystyle \{1, 2, \dots, 90\}. For any set of 3 distinct numbers, there are 3!=6\displaystyle 3! = 6 possible permutations of the order in which they could be drawn: 1. X1<X2<X3\displaystyle X_1 < X_2 < X_3 (ascending order) 2. X1<X3<X2\displaystyle X_1 < X_3 < X_2 3. X2<X1<X3\displaystyle X_2 < X_1 < X_3 4. X2<X3<X1\displaystyle X_2 < X_3 < X_1 5. X3<X1<X2\displaystyle X_3 < X_1 < X_2 6. X3<X2<X1\displaystyle X_3 < X_2 < X_1 (descending order) Since each chip is drawn with equal probability, all 6 permutations are equally likely. Out of these 6 permutations, exactly 1 is in descending order (namely X3<X2<X1\displaystyle X_3 < X_2 < X_1, which corresponds to the first chip being the largest, the second chip being the second largest, and the third chip being the smallest). Therefore, the probability that the numbers are drawn in descending order is: Probability=16\displaystyle \text{Probability} = \frac{1}{6}. Hence, **Option C** 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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