let's play a game · Physics Brawl 2018 #FoL.3
Physics · mechanics
Problem
You are playing a game which consists of separate rounds. When you lose a round, you lose a point. When you win a round, you gain a point, and if you have also won the previous two rounds, you gain another point as a bonus. If the chance of winning a round is 50% (draw is not possible), what is the long-term average (expected value) of the number of points gained per round?Intuitive explanation
This is a problem about expected value in a system with state-dependent rewards—the key insight is that you can't just average the point outcomes; you need to account for how often you're actually in each situation. Set up states based on your recent win history (like "just lost," "won once," "won twice in a row"), find the long-run probability of being in each state, then weight each state's expected points by how often you occupy it. The bonus rule creates a dependency that makes the naive 50% approach miss the real pattern.
Intuitive explanation
This is a problem about expected value in a system with state-dependent rewards—the key insight is that you can't just average the point outcomes; you need to account for how often you're actually in each situation. Set up states based on your recent win history (like "just lost," "won once," "won twice in a row"), find the long-run probability of being in each state, then weight each state's expected points by how often you occupy it. The bonus rule creates a dependency that makes the naive 50% approach miss the real pattern.