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Boston Consulting Group·Data Scientist·Online Assessment (OA)·Intermediate

Intermediate
May 2026

Summary

BCG's CodeSignal assessment for a Data Scientist role had a quick probability section that felt more like a stats refresher than anything tricky. Two questions, back to back, both classical.

Questions Asked (2)

Q1

You flip a fair coin three times. What is the probability of getting exactly two heads?

Algorithms & Data Structures
Author's notes

Binomial setup, pretty clean.

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AI HintsAI Generated

Suggested Approach

Start by clearly defining the sample space of 8 equally likely outcomes for three coin flips. Then count the number of outcomes with exactly two heads (3) and divide by the total (8) to get 3/8. Optionally, mention the binomial formula as a generalization.

Pro tip: After giving the answer, briefly connect it to a real-world data science scenario, such as estimating conversion rates or A/B testing, to show practical insight. Also, explicitly state that the coin is fair and flips are independent, as this assumption is key.

1. Define the experiment and assumptions

State that each flip is independent and has two equally likely outcomes (H or T). Clarify that we want exactly two heads in three flips.

2. Enumerate the sample space

List all 2^3 = 8 possible sequences: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Emphasize they are equally likely.

3. Count favorable outcomes

Identify the sequences with exactly two heads: HHT, HTH, THH. There are 3 such outcomes.

4. Compute probability

Divide the number of favorable outcomes by the total number of outcomes: 3/8 = 0.375 or 37.5%.

5. Generalize (optional)

Mention the binomial probability formula: P(X=k) = C(n,k) * p^k * (1-p)^(n-k). For n=3, k=2, p=0.5, this gives C(3,2)*(0.5)^2*(0.5)^1 = 3/8.

Key Points to Mention

  • The sample space consists of 8 equally likely outcomes.
  • There are exactly 3 outcomes with two heads: HHT, HTH, THH.
  • The probability is 3/8 or 0.375.
  • The binomial distribution can be used to generalize this calculation.
  • Independence and fairness of the coin are key assumptions.
  • This type of probability calculation is fundamental in data science for modeling binary outcomes.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.

Q2

A bag has 4 red and 6 blue balls. If you draw two balls without replacement, what is the probability that both are red?

Algorithms & Data Structures
Author's notes

Without replacement tripped me up for half a second.

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AI HintsAI Generated

Suggested Approach

Clarify that the draws are without replacement, then compute the probability using the multiplication rule for dependent events: P(first red) × P(second red | first red). Alternatively, use combinations to count favorable outcomes over total outcomes.

Pro tip: State your assumptions explicitly (e.g., each ball is equally likely to be drawn) and mention that you can verify the result with a quick simulation or by checking the complement, showing rigor and practical data science intuition.

1. Identify the total number of balls and the number of red balls

There are 4 red and 6 blue balls, so the total is 10 balls. The number of red balls is 4.

2. Compute the probability that the first ball is red

Since there are 4 red balls out of 10, P(first red) = 4/10 = 2/5.

3. Compute the conditional probability that the second ball is red given the first was red

After drawing a red ball, there are 3 red balls left out of 9 total. So P(second red | first red) = 3/9 = 1/3.

4. Multiply the probabilities to get the final answer

P(both red) = (4/10) × (3/9) = 12/90 = 2/15 ≈ 0.1333.

5. Optionally, verify using combinations

Number of ways to choose 2 red from 4 is C(4,2)=6. Total ways to choose 2 from 10 is C(10,2)=45. Probability = 6/45 = 2/15.

Key Points to Mention

  • Without replacement means the draws are dependent; the probability changes after the first draw.
  • Use the multiplication rule: P(A and B) = P(A) × P(B|A).
  • Alternatively, use combinations: favorable outcomes over total outcomes.
  • The final probability is 2/15 or approximately 0.1333.
  • State assumptions: all balls are equally likely to be drawn, and draws are random.
  • Mention that this is a hypergeometric distribution scenario (sampling without replacement from a finite population).

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.