Solve probability and expectation problems

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?