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Solve core probability/statistics mini-problems

UpstartData Scientist · Onsite
MediumData ScientistOnsiteStatistics & Math

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Category: Statistics & Math; this prompt evaluates core probability and statistical concepts — exponential decay and discrete survival counts, Bayesian updating, properties of population OLS coefficients, conditional-probability reasoning exemplified by Monty Hall, the coupon-collector expected time, and the conceptual distinction between likelihood and probability — relevant for Data Scientist roles. It is commonly asked because it probes foundational distributional intuition, independence and conditional inference, and expectation/estimation at an introductory-to-intermediate theoretical level that underpins applied modeling and experimental interpretation.

Answer the following probability/statistics interview questions. Assume all randomness is independent unless stated otherwise.

  1. Radioactive decay (half-life): A radioactive atom has half-life = 1 day. You start with n = 100 identical atoms.
  • (a) What is the probability a given atom is still undecayed after m = 10 days?
  • (b) What is the distribution of the number of atoms still undecayed after 10 days?
  • (c) Compute the expected number of atoms remaining after 10 days, and the probability that at least one atom remains.

  1. Bayes’ rule (generic form): Let event A be the “true condition” and event B be an observed test result. You are given P(A), P(B\mid A), and P(B\mid A^c). Derive P(A\mid B).

  2. OLS coefficients in two regressions: Let (y = x + e) where (x \sim \mathcal N(0,1)), (e \sim \mathcal N(0,1)), and (x) and (e) are independent.

  • (a) In the population OLS regression of (y) on (x) (with intercept), what is the slope coefficient?
  • (b) In the population OLS regression of (x) on (y) (with intercept), what is the slope coefficient?

  1. Monty Hall: You pick 1 of 3 doors. The host, who knows where the prize is, opens a different door showing no prize, then offers you the chance to switch to the remaining closed door. What strategy maximizes your win probability, and what is that probability?

  2. n-sided die / coupon collector: You repeatedly roll a fair n-sided die. What is the expected number of rolls required to have seen every face at least once?

  3. Likelihood: In parametric modeling, explain what a likelihood is and how it differs from a probability statement.