Login

u/Lead_balloon784

▲ 3 r/CMI

CMI MSc DS Incorrect Solution in Answer key (Part A, Q6)

Objection to Answer Key: Part (A), Question 6

CMI M.Sc. Data Science Entrance Examination 2026

The Question

A box contains 3 red balls and 2 blue balls. Two balls are drawn one after another without replacement. Let A be the event that both balls are red, and B the event that the first ball is red. Which of the following are true?

  • (a) P(A) = 3/10
  • (b) P(B) = 3/5
  • (c) P(Bᶜ | Aᶜ) = 2/3 (here Xᶜ is the event that X does not take place)
  • (d) P(A | B) = 3/5

Official Key Answer: (a), (b), (c)
Correct Answer: (a), (b)

Correction for Option (c)

The official solution states: "Note that Aᶜ = {RB, BB, BR} so there are exactly two possibilities for the first ball being blue. Hence c is true." However, this logic falsely assumes outcomes in Aᶜ to be equiprobable. Because sampling is performed without replacement, the exact probabilities for the constituent states are asymmetric:

P(RB) = 3/5 × 2/4 = 3/10
P(BR) = 2/5 × 3/4 = 3/10
P(BB) = 2/5 × 1/4 = 1/10

By definition, Bᶜ (the first ball is blue) is a strict subset of Aᶜ (not both balls are red), meaning Bᶜ ∩ Aᶜ = Bᶜ. Evaluating the conditional probability equation yields:

P(Aᶜ) = 1 - P(A) = 1 - 3/10 = 7/10
P(Bᶜ ∩ Aᶜ) = P(Bᶜ) = P(BR) + P(BB) = 3/10 + 1/10 = 4/10

P(Bᶜ | Aᶜ) = P(Bᶜ ∩ Aᶜ) / P(Aᶜ) = (4/10) / (7/10) = 4/7

Since 4/7 (≈ 0.5714) ≠ 2/3, option (c) is false.

Conclusion

The correct set of true statements is (a) and (b). I request the admissions committee to re-evaluate the scoring template for this question.

#Update: They've corrected it now

reddit.com
u/Lead_balloon784 — 3 days ago
▲ 3 r/CMI

Saw somebody asking for it, thought I'd make a post for better visibility. Feel free to add to it if you remember any

Questions I remember incase someone wants:
Part A:
Q1: Let n = 1. If at any point you can either replace n by 2n, or 2n+1 which of the following numbers can be, reached? (I dont remember the exact numbers but all are correct since all positive integers are reachable)
Q2: Dont remember the exact wording. But it was a simple derangement problem. Answer is D5 = 44.
Q3: Which of these are true at all points where they are defined? |sinx|<|x|, |tanx|>|x|, and some other options i dont remember
Q4: Consider a function from {1234}->{123}. A] Find number of functions with exactly 2 images, B] number of onto functions, C] If f(1) = 1 and f(2) = 2, find number of onto functions, and some more things i dont remember.
Q5: A jar has 3 red balls and 2 blue balls. Two balls are picked without replacement. Let B be event that first ball is red. A be event that both balls are red. Find P(A), P(B), P(A/B), P(B'/A')
Q6: Consider a switchboard with all switches off. For numbers i running from 1 through 10, toggle kth switch if the i divides the k. Find which switches will be turned on after going through all 10 numbers (I think answer was 1,4,9)
Q7-9: Psuedocode Enigma. I remember but too lazy to write the whole problem out

Part B:
Q1: Given f(x+y) = f(x) × f(y). Prove that if f(0) = 1 then f(r) = f(1)^r for any rational number r.
Q2: If a prime number p, divides n^2+n+1, find (n^3) % p.
Q3: If a prime number p, divides n^2+n+1 and p > 3, show that 1, n and n^2 all have distinct remainders when divided by p.
Q4. Sum of first 100 numbers which are not multiples of 3 or 7.
Q5: Given A^2 = A for some 2x2 matrix A, Find Trace(A).
Q6: T = max{x1, x2, x3) where xi follows Uniform(0, theta). Find pdf, cdf, mean and variance of T.
Q7: Consider a sequence of n natural numbers {1, 2, .... , n}. Suppose they are randomly permuted. A position "i" is considered maxima if it is greater than all numbers to left of it. Eg: 1,3,5,2. Here 1,3,5 are all maximas. Find the probability that in a sequence of length n, the ith position will be a maxima.
Q8: Find the expected number of maxima in a sequence of length n.
Q9: The faulty llm model question. I dont remember the exact statements.
Q10: Find the conditions under which the sum of first n natural numbers will divide the sum of squares of first n natural numbers.
Q11: Pseudocode. Function Steps(whille n>1, if n is odd n = n-1 and count=count+1, if n is even n = n/2count=count+1, return count). Find steps(16) or some other number i dont remember
Q12: Pseudocode. Function Steps(whille n>1, if n is odd n = n-1 and count=count+1, if n is even n = n/2count=count+1). For a number which is k digits long in binary find the max value steps() can return

reddit.com
u/Lead_balloon784 — 18 days ago