Solve probability and expectation problems

Solve the following probability and mathematical interview problems:

1. Squid Game glass bridge: The bridge consists of B steps in sequence. At each step there are two glass panels, and exactly one is safe. P players attempt the crossing one after another. When a step has not yet been revealed, the player at that step picks left or right with equal probability; after the safe panel has been revealed, every later player can rely on that information. Give a formula for the probability that the i-th player in line makes it all the way across. Also give the probability that at least one of the P players survives.
2. Sum of two uniforms: Suppose $X, Y \overset{iid}{\sim} \text{Uniform}(0,1)$. Find the PDF of $S = X + Y$.
3. Coupon collector with a die: Roll a fair six-sided die until each of the six faces has shown up at least once. What is the expected number of rolls?
4. Variance of a coordinate on the unit sphere: A point is chosen uniformly from the surface of the 3D unit sphere $x^2 + y^2 + z^2 = 1$, with coordinates $(X,Y,Z)$. Find $\mathrm{Var}(X)$.
5. Sequentially keep k draws: You observe $n$ iid draws from $\text{Uniform}(0,1)$ one by one. After each draw, you must decide immediately and irrevocably 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?

Overview: These problems test probabilistic reasoning and expectation skills in both discrete and continuous settings, including conditional probability with evolving information, convolution and distribution derivation, occupancy/collector problems, symmetry-based variance on manifolds, and sequential selection/optimal stopping.