Probability You Picked the Strongest Cat Given Your Cat Lost the Race

Three cats run a race, and exactly one of them wins. Their probabilities of winning are $x$, $y$ and $z$, with $x + y + z = 1$. Before the race you pick one of the three cats uniformly at random to cheer for. After the race you learn that the cat you picked did not win. What is the probability that you picked the strongest cat, the one with the highest probability of winning? The assessment varies this template: the condition may instead be that your cat won, the question may ask about the weakest cat instead of the strongest, and the contestants may be players rather than cats. Give the answer for each of the four combinations of condition and target. The specific probabilities are generated randomly, and the required rounding is stated with each question. ```hint List the ways it can happen Split the observed event into one case per cat you might have picked, and weigh each case before you condition. ``` ### Constraints and Clarifications - $x$, $y$ and $z$ are nonnegative and sum to 1. - Your pick is independent of how the race turns out. - The strongest and the weakest cats are unique. ### Clarifying Questions - Is there always exactly one winner, so that the three probabilities sum to one? - If two cats share the highest winning probability, which one counts as the strongest? - Is the pick uniform, or could it depend on the cats' odds? ### What a Strong Answer Covers - The joint probability of each possible pick together with the observed outcome - The probability of the conditioning event, and how it simplifies when the winning probabilities sum to one - All four variants (won or lost, strongest or weakest) obtained from one general expression - Sanity checks against extreme cases, such as three equally strong cats or a cat that never loses ### Follow-up Questions - If each cat's success were a separate event, so that zero or several cats could succeed, which parts of your answer would still hold? - If you picked a cat with probability proportional to its chance of winning instead of uniformly, what would the answer be? - Given that your cat lost, what is the probability that the strongest cat won the race?