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Capital One·Data Scientist·Technical Phone Screen·Intermediate

Intermediate
Jun 2026

Summary

Capital One data scientist interview with some solid probability questions that made me feel like I was back in a stats class I barely passed. Two-part problem format, both requiring actual numeric answers with shown work.

Questions Asked (2)

Q1

You have a box with 3 fair coins and 2 double-headed coins. You pick one at random, flip it three times, and get heads all three times. What is the probability the coin you picked is double-headed?

Product Analytics & MetricsAlgorithms & Data Structures
Author's notes

Bayesian updating, classic setup.

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AI HintsAI Generated

Suggested Approach

Recognize this as a conditional probability problem and apply Bayes' theorem. Clearly define the events, compute the prior probabilities and the likelihood of observing three heads given each coin type, then calculate the posterior probability that the chosen coin is double-headed.

Pro tip: State your assumptions explicitly (e.g., fair coins have P(H)=0.5, double-headed have P(H)=1, flips are independent) and verify that the posterior is intuitive: three heads strongly suggests the double-headed coin, so the probability should be high.

1. Define events and priors

Let D be the event that the chosen coin is double-headed, and F be the event that it is fair. Compute the prior probabilities: P(D)=2/5, P(F)=3/5.

2. Determine likelihoods

Compute the probability of observing three heads in a row given each coin type: P(3H|D)=1^3=1, P(3H|F)=(1/2)^3=1/8.

3. Apply Bayes' theorem

Use Bayes' theorem to find P(D|3H) = [P(3H|D)*P(D)] / [P(3H|D)*P(D) + P(3H|F)*P(F)].

4. Calculate the posterior

Plug in the numbers: numerator = 1*(2/5)=2/5; denominator = 2/5 + (1/8)*(3/5)=2/5+3/40=16/40+3/40=19/40. So P(D|3H) = (2/5)/(19/40) = (16/40)/(19/40)=16/19.

5. Interpret the result

Conclude that the probability is approximately 0.842, meaning there is an 84.2% chance the coin is double-headed given three heads.

Key Points to Mention

  • Bayes' theorem and conditional probability
  • Prior probabilities based on the composition of the box
  • Likelihood of observing three heads for each coin type
  • Independence of coin flips
  • The final answer 16/19 or approximately 84.2%
  • Sanity check: the result is greater than the prior 2/5, as expected

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

Q2

An urn has 5 red, 4 blue, and 3 green balls. You draw 3 without replacement. What is the probability of getting exactly two distinct colors where at least one ball is red?

Algorithms & Data StructuresProduct Analytics & Metrics
Author's notes

This one took me longer than it should have.

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AI HintsAI Generated

Suggested Approach

Break down the event into mutually exclusive cases based on the color composition of the three balls. Calculate the probability for each case using combinations and sum them, ensuring the condition 'exactly two distinct colors and at least one red' is satisfied.

Pro tip: After computing the probability, verify by considering the complement or using a quick simulation to catch errors. Also, mention that this type of combinatorial probability is common in product analytics for estimating user behavior patterns.

1. Understand the event

Identify that we need exactly two distinct colors among the three balls, and at least one ball must be red. This means the color composition can be either (Red, Blue, Blue) or (Red, Green, Green) or (Red, Red, Blue) or (Red, Red, Green).

2. Enumerate valid color compositions

List all possible color multisets of size 3 that have exactly two distinct colors and include red. These are: 2 Red + 1 Blue, 2 Red + 1 Green, 1 Red + 2 Blue, 1 Red + 2 Green.

3. Compute combinations for each case

For each composition, calculate the number of ways to choose the balls from the urn using combinations. For example, for 2 Red + 1 Blue: C(5,2)*C(4,1).

4. Sum favorable outcomes and divide by total

Add the counts from all valid cases to get the total favorable outcomes. Divide by the total number of ways to draw 3 balls from 12 (C(12,3)) to get the probability.

5. Simplify and verify

Simplify the fraction if possible. Optionally, verify by computing the probability via an alternative method (e.g., sequential draws) or by checking that the sum of probabilities of all disjoint events equals 1.

Key Points to Mention

  • Use of combinations (nCr) to count outcomes without replacement.
  • Mutually exclusive cases for exactly two distinct colors with at least one red.
  • Total number of possible draws is C(12,3) = 220.
  • Favorable outcomes: C(5,2)*C(4,1) + C(5,2)*C(3,1) + C(5,1)*C(4,2) + C(5,1)*C(3,2) = 40 + 30 + 30 + 15 = 115.
  • Probability = 115/220 = 23/44 ≈ 0.5227.
  • Cross-check with complement or simulation for accuracy.

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