Mavensecurities·Data Scientist·Technical Phone Screen·Intermediate

Intermediate

Jul 2026

Summary

Interviewed for a Data Scientist role at Mavensecurities and ran into a combinatorics probability question that felt more like a math competition problem than anything I'd prepped for.

Questions Asked (1)

You randomly draw 3 distinct numbers from 1 to 30 without replacement, where all combinations are equally likely. What is the probability that the three numbers can form an arithmetic progression?

Algorithms & Data Structures

Author's notes

Spent way too long second-guessing myself on how to count valid arithmetic progressions.

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AI HintsAI Generated

Suggested Approach

First, clarify that the three numbers must be distinct and drawn without replacement, so order doesn't matter. Then, count the total number of possible triples (C(30,3)) and the number of triples that form an arithmetic progression. Finally, compute the probability as the ratio of favorable outcomes to total outcomes.

Pro tip: Mention that the common difference must be an integer and that the middle term determines the progression. This shows you understand the structure and can avoid double-counting.

1. Understand the problem

Confirm that we are drawing 3 distinct numbers from 1 to 30 without replacement, and that any set of 3 numbers is equally likely. The order does not matter.

2. Count total outcomes

Calculate the total number of ways to choose 3 numbers from 30, which is C(30,3) = 4060.

3. Count favorable outcomes

Count the number of 3-element subsets that form an arithmetic progression. For a progression a, a+d, a+2d with d ≥ 1, the largest term a+2d ≤ 30. For each d from 1 to 14, count the number of valid a: a can be 1 to 30-2d, so there are 30-2d choices. Sum over d: Σ_{d=1}^{14} (30-2d) = 14*30 - 2*(14*15/2) = 420 - 210 = 210.

4. Compute probability

Divide the number of favorable outcomes (210) by the total number of outcomes (4060) to get the probability. Simplify the fraction if possible: 210/4060 = 21/406 = 3/58.

5. Verify and present

Double-check the count by considering alternative methods (e.g., fixing the middle term) to ensure no mistakes. Present the final answer clearly: 3/58 ≈ 0.0517.

Key Points to Mention

The total number of outcomes is C(30,3) = 4060.
An arithmetic progression is determined by its first term and common difference, with the constraint that all terms are within 1 to 30.
The common difference d must be a positive integer, and the maximum d is 14 because the smallest progression with d=14 is 1,15,29 and with d=15 would be 1,16,31 (out of range).
For each d, the number of valid progressions is 30 - 2d, leading to a sum of 210.
The probability simplifies to 3/58, which is approximately 0.0517 or 5.17%.
Mention that order does not matter, so we count combinations, not permutations.

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