Solve core probability/statistics mini-problems
Quick Overview
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.
-
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.
- 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) .
-
OLS coefficients in two regressions:
Let
where
,
, and
and
are independent.
- (a) In the population OLS regression of on (with intercept), what is the slope coefficient?
- (b) In the population OLS regression of on (with intercept), what is the slope coefficient?
- 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?
- 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 ?
- Likelihood: In parametric modeling, explain what a likelihood is and how it differs from a probability statement.