You are asked to solve the following probability and mathematical interview problems: 1. **Squid Game glass bridge**: There are **B** sequential bridge steps. Each step has two glass panels, exactly one of which is safe. **P** players cross in order. Whenever a step has not been revealed yet, the player facing it chooses left/right uniformly at random; once the safe panel is revealed, all later players can use that information. Derive a formula for the probability that the **i-th player in line** successfully crosses the bridge. Also state the probability that **at least one of the P players** survives. 2. **Sum of two uniforms**: Let \(X, Y \overset{iid}{\sim} \text{Uniform}(0,1)\). Find the PDF of \(S = X + Y\). 3. **Coupon collector with a die**: You roll a fair six-sided die until all six faces have appeared at least once. What is the expected number of rolls? 4. **Variance of a coordinate on the unit sphere**: A point is sampled uniformly from the **surface** of the 3D unit sphere \(x^2 + y^2 + z^2 = 1\), and its coordinates are \((X,Y,Z)\). Find \(\mathrm{Var}(X)\). 5. **Sequentially keep k draws**: You observe \(n\) iid draws from \(\text{Uniform}(0,1)\) one at a time. After each draw, you must immediately and irrevocably decide whether to keep it or discard it, and by the end you must have kept exactly \(k\) draws. What is the optimal strategy, and how can you compute the optimal expected total value?