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CAT and MBA Entrance Preparation Module 6

Quantitative Ability: geometry, counting and probability interview questions

Interview questions on “Quantitative Ability: geometry, counting and probability”: warm-ups, then what each harder question tests and how a good answer starts.

7 questions: 1 warm-up and 6 with model answers · Levels: Fresher to mid

Warm-up questions

Short questions with short answers, each answered in full.

Fresher

What is a derangement?

Answered in the lesson Permutations and combinations: CAT methods

Questions with model answers

What each question tests, the first step of a good answer and the follow-ups to expect.

Fresher to mid

Seven people sit at a round table. Asha and Bala refuse to sit side by side, while Chitra and Dev insist on sitting together. How many seatings are possible, and how would you check the count?
What it tests

Gluing a pair at a round table, removing the seatings that break a second rule, and checking a count by a second route.

First step

Glue Chitra and Dev into one unit with 2 orders inside: six units sit round the table in 5! ways, so 240 seatings keep them together. From these, take away the seatings in which Asha and Bala also sit side by side.
Follow-ups
- How does the count change for a row of seven chairs?
- How many seatings keep Asha and Bala apart if Chitra and Dev may sit anywhere?
- How would a listing of the seatings check the answer?
From the lesson Permutations and combinations: CAT methods
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In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 3 minutes
Fresher to mid

Fifteen identical sweets are shared among four children so that every child gets at least 2 and at most 6. How many shares are possible, and how would you reach the number quickly in a test?
What it tests

Turning a lower limit into a fresh start, and handling an upper limit by subtracting and adding back with stars and bars.

First step

Give every child 2 sweets first. That leaves 7 sweets for 4 children, each now allowed at most 4 more. Count the shares of 7 with no cap by stars and bars, then remove the shares in which some child takes 5 or more.
Follow-ups
- What is the count with no upper limit?
- What if the children are identical, so that only the sizes of the shares matter?
- What changes if the cap is 5 instead of 6?
From the lesson Permutations and combinations: CAT methods
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 3 minutes
Fresher to mid

Six different prizes go to four students so that every student gets at least one. A classmate counts 6 × 5 × 4 × 3 ways to give each student a first prize, then 4² ways for the other two: 5,760. Is that right? Find the count.
What it tests

Counting distinct items into distinct boxes with none empty by inclusion–exclusion, and finding why a first-prize method reaches a share more than once.

First step

Start from all 4⁶ = 4,096 ways, in which each prize picks a student. Take away the ways that leave a named student out, 3⁶ for each of the 4 students, then add back the ways that leave two out and take away those that leave three out.
Follow-ups
- How many times does the classmate's method reach a share in which one student gets three prizes?
- What is the count if the six prizes are identical?
- How many shares give every student at most two prizes?
From the lesson Permutations and combinations: CAT methods
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A fair die is rolled until the first six appears. What is the probability that more than three rolls are needed, and how many rolls are needed on average?
What it tests

Turning "more than k tries" into "no success in the first k", and finding an expected count for repeated independent tries.

First step

More than three rolls are needed exactly when none of the first three rolls is a six. Each roll misses with probability 5/6, whatever the others did, so multiply the three chances.
Follow-ups
- What is the chance that the first six comes on an even-numbered roll?
- How many rolls give a better than even chance of at least one six?
- On average, how many rolls are needed to see two sixes in a row?
From the lesson Probability for CAT: counting, Bayes, expectation
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

One per cent of the components a supplier ships are faulty. A test flags 95 % of faulty components and 4 % of good ones. A component is flagged. How likely is it to be faulty, and what would you do next?
What it tests

Bayes' rule with a small base rate, reading the answer as counts, and judging what a second piece of evidence is worth.

First step

Think of 10,000 components: 100 are faulty and 95 of them are flagged, while 9,900 are good and 396 of them are flagged too. Compare the flagged faulty components with all the flagged components.
Follow-ups
- A second, independent test also flags the component. Now what?
- How low must the false-alarm rate be for a flag to mean an even chance of a fault?
- Why is 95 % the wrong answer?
From the lesson Probability for CAT: counting, Bayes, expectation
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A stall charges ₹50 for a roll of two dice and pays ₹10 times the larger number shown. Is the game worth playing, and what fee would make it fair?
What it tests

Building the distribution of a maximum by counting, and turning it into an expected value and a fair fee.

First step

Count the outcomes in which the larger number is exactly m: the outcomes with both dice at most m, m² of them, less those with both at most m − 1. That gives 1, 3, 5, 7, 9 and 11 outcomes for m = 1 to 6.
Follow-ups
- What is the chance of winning money on one play?
- How would the answer change if the stall paid ₹10 times the total instead?
- How many plays does the stall need to expect a profit of ₹1,000?
From the lesson Probability for CAT: counting, Bayes, expectation
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes

All CAT and MBA Entrance Preparation interview questions · The CAT and MBA Entrance Preparation track

Warm-up questions

What is a derangement?

Questions with model answers

Seven people sit at a round table. Asha and Bala refuse to sit side by side, while Chitra and Dev insist on sitting together. How many seatings are possible, and how would you check the count?

Fifteen identical sweets are shared among four children so that every child gets at least 2 and at most 6. How many shares are possible, and how would you reach the number quickly in a test?

Six different prizes go to four students so that every student gets at least one. A classmate counts 6 × 5 × 4 × 3 ways to give each student a first prize, then 4² ways for the other two: 5,760. Is that right? Find the count.

A fair die is rolled until the first six appears. What is the probability that more than three rolls are needed, and how many rolls are needed on average?

One per cent of the components a supplier ships are faulty. A test flags 95 % of faulty components and 4 % of good ones. A component is flagged. How likely is it to be faulty, and what would you do next?

A stall charges ₹50 for a roll of two dice and pays ₹10 times the larger number shown. Is the game worth playing, and what fee would make it fair?