Calculating Ken’s Expected Coin Flip Payout

This question is about a simple but insightful expected value calculation under a cap on the payout. Even though the setup uses basic coin flips and powers of two, it hides a classic structure behind many payoff distributions in quantitative finance. It is a clean, interview-friendly way to link discrete probability with payoffs that cannot grow indefinitely, which mirrors many products in markets.

For quant prep, it trains comfort with expected value, geometric distributions, and understanding how imposing a hard limit on a random payoff changes the distribution. It reinforces working systematically with infinite scenarios that effectively truncate, and builds intuition for bounded versus unbounded risk and reward.

This matters in quant interviews because many derivative payoffs, risk limits, and trading strategies have caps or floors. Being fluent with capped expectations signals strong probabilistic thinking, vital for pricing, risk, and algorithmic trading roles.

What it tests

Whenever a random payout is capped at a maximum value, the expected value calculation splits naturally into two regions: outcomes below the cap, where the payout function is unaltered, and outcomes above the cap, where the payout is constant at the cap. The key is that the expectation over a capped random variable is the sum of the uncapped expected values up to the threshold, plus the cap times the probability of exceeding the threshold. This structure arises because the minimum (or maximum) of a function and a constant partitions the sample space. The underlying reason is that expectation is linear, so you can sum over disjoint events ("below the cap" and "at the cap") and weight each by its probability. This approach generalizes to any situation where a random process is stopped or bounded by a fixed limit, not just coin flips or powers of two.