Sampling Strategy and Expected Value

Problem A probability puzzle on sampling strategy: decide whether one strategy beats another for winning, given that the average-case outcomes are expected to be equal.

Be ready to discuss

• Setting up the sample space and the random variable precisely before computing anything - what exactly counts as a "win".
• Computing expected value by linearity of expectation, and why linearity holds even when the underlying events are dependent.
• Showing two strategies are equivalent in expectation while their variances differ, so "same average" does not mean "same chance of winning".
• Symmetry and exchangeability arguments that let you conclude equality without grinding through the algebra.
• Conditional expectation and the law of total expectation for strategies that adapt based on what has been sampled so far.
• Why maximising expected value is the wrong objective when the goal is P(win) - a higher-variance strategy can be correct when you must beat a fixed threshold.
• Sanity-checking the result on a small case (n = 2 or 3) by enumerating outcomes exhaustively.