Pick your threshold

Optimizing a threshold-choice problem under discrete probability

This is a straightforward but instructive probability-optimization question. You must choose a single parameter before observing an outcome, then compare the expected payoff under each choice. It tests whether you can enumerate the sample space, compute probabilities correctly, and systematically optimize a discrete decision variable.

The key insight is that increasing your threshold raises the payout when you win, but lowers the probability of winning. Your task is to find the sweet spot—the value of k that balances these competing forces to maximize expected value. A systematic approach is to compute expected payout for each candidate threshold, then select the maximum.

• Probability mass functions over discrete outcomes
• Expected value as a function of a decision parameter
• Trade-off between payoff magnitude and success probability