Virtu Financial·Data Scientist·Technical Phone Screen·Intermediate

IntermediatePrefer not to say

May 2026

Summary

Virtu Financial data scientist interview with some pretty gnarly probability questions. The urn problem showed up and it was more involved than it looked on the surface, especially the second part where they change the ball counts and want both an intuitive guess and the exact math.

Questions Asked (2)

An urn has 5 red, 5 blue, and 5 green balls. You draw one at a time uniformly at random. Green balls go back immediately; red and blue are kept. You stop when you're holding 3 balls. What's the probability that at least two of the three kept balls are red?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The green ball mechanic is a bit of a distraction.

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AI HintsAI Generated

Suggested Approach

Model the process as a Markov chain where the state is the number of red and blue balls kept so far, and green draws are self-loops. Compute the probability of reaching a state with at least two red balls when the total kept reaches 3, using symmetry and dynamic programming or recursive equations.

Pro tip: Leverage the symmetry between red and blue to simplify calculations, and consider using a recursive approach with memoization to handle the varying probabilities as balls are removed.

1. Define the state space

Identify the relevant states as (r, b), the number of red and blue balls currently kept, with r + b ≤ 3. Green draws do not change the state.

2. Determine transition probabilities

At each state, compute the probability of drawing red, blue, or green based on the remaining balls in the urn. Green returns immediately, so it only affects the waiting time.

3. Set up recursive equations

Let P(r, b) be the probability of eventually having at least two red balls when stopping, given current state (r, b). Write equations for P(r, b) in terms of P(r+1, b), P(r, b+1), and P(r, b) itself (due to green).

4. Solve the equations

Solve the recursive equations, using boundary conditions: if r ≥ 2 and r + b = 3, P = 1; if r < 2 and r + b = 3, P = 0. Use symmetry to reduce the number of equations.

5. Compute the final probability

Evaluate P(0,0) to get the desired probability. Verify the result by considering alternative methods or simulation if time permits.

Key Points to Mention

Markov property: the future depends only on the current counts of red and blue balls kept.
Green balls act as self-loops and can be ignored for the state transitions, but they affect the total number of draws.
Symmetry between red and blue simplifies the state space and equations.
The process stops when 3 non-green balls are kept, so the total number of red and blue balls drawn is exactly 3.
Use of dynamic programming or recursive equations to compute probabilities.
Boundary conditions: absorbing states when r + b = 3.

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

Now the urn has 10 red, 5 blue, and 5 green balls, same process. Give an intuitive estimate first, then compute the exact probability that at least two of the three held balls are red.

Algorithms & Data StructuresProduct Analytics & Metrics

Author's notes

The intuition part is what tripped me up a little.

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AI HintsAI Generated

Suggested Approach

First, give a quick intuitive estimate by reasoning about the expected number of red balls and the probability of drawing at least two reds. Then, compute the exact probability using combinatorics: count the total ways to choose 3 balls from 20, and count the favorable outcomes (exactly 2 reds or exactly 3 reds). Finally, express the probability as a fraction or decimal.

Pro tip: Show your intuition first to demonstrate probabilistic thinking, then verify with exact calculation. Mention that the hypergeometric distribution applies here, which is common in quantitative finance interviews.

1. Intuitive Estimate

Reason that the expected number of red balls in 3 draws is 1.5, so getting at least 2 reds is plausible but not highly likely. Estimate the probability to be around 20-30%.

2. Define the Sample Space

Calculate the total number of ways to choose 3 balls from 20 without replacement: C(20,3) = 1140.

3. Count Favorable Outcomes

Count the number of ways to get exactly 2 reds and exactly 3 reds. For exactly 2 reds: C(10,2)*C(10,1) = 450. For exactly 3 reds: C(10,3) = 120. Total favorable = 570.

4. Compute Probability

Divide favorable outcomes by total outcomes: 570/1140 = 0.5. So the exact probability is 50%.

5. Compare and Reflect

Compare the exact result (50%) with your intuitive estimate. If they differ, discuss why (e.g., intuition may underestimate due to the high proportion of red balls).

Key Points to Mention

Hypergeometric distribution: drawing without replacement from a finite population.
Combinatorial calculation: using combinations to count outcomes.
Expected value: expected number of red balls is 1.5, which informs intuition.
Complementary counting: sometimes easier to count the complement (0 or 1 red) but here direct counting is straightforward.
Assumption of random sampling: each subset of 3 balls is equally likely.
Practical relevance: such probability calculations are common in quantitative finance and risk assessment.

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