Solve core probability/statistics mini-problems

Work through the probability and statistics interview questions below. Treat all random quantities as independent unless a question states otherwise.

1. Radioactive decay (half-life): A radioactive atom has a half-life of 1 day. You begin with n = 100 identical atoms.

   • (a) What is the probability that a given atom has not decayed 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.

2. Bayes’ rule (generic form): Take event A to be the “true condition” and event B to be an observed test result. Given $P(A)$, $P(B\mid A)$, and $P(B\mid A^c)$, derive $P(A\mid B)$.

3. 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?

4. Monty Hall: You choose one of three doors. The host, who knows where the prize is, opens a different door with no prize behind it and then offers you the chance to switch to the remaining unopened door. Which strategy maximizes your win probability, and what is that probability?

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

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

Overview: Category: Statistics & Math; this question set tests core probability and statistical concepts — exponential decay and discrete survival counts, Bayesian updating, population OLS slope behavior, conditional-probability reasoning through Monty Hall, coupon-collector expected waiting time, and the conceptual difference between likelihood and probability — relevant for Data Scientist roles. It is commonly asked because it checks foundational distributional intuition, independence and conditional inference, and expectation/estimation at an introductory-to-intermediate theoretical level that underpins applied modeling and experimental interpretation.

This question came from a data scientist interview experience.