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. 2) **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)**. 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 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? 5) **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**? 6) **Likelihood:** In parametric modeling, explain what a **likelihood** is and how it differs from a probability statement.