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Puzzling

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    $\begingroup$ That's why in probability puzzles people always repeat "independently uniformly random". $\endgroup$
    – WhatsUp
    Commented Dec 14, 2021 at 14:28
  • 6
    $\begingroup$ But could you deduce from those three scenarios that A is most likely to win if you combine the results? A is always expected to win half the races in every scenario, whereas B and C are expected to win less often (Once half the races and once less than half) $\endgroup$
    – QBrute
    Commented Dec 15, 2021 at 11:38
  • 4
    $\begingroup$ @TheRubberDuck You are confounding probability and statistics, a common mistake. Whatever your reasons to believe scenario 3 is more likely I'd be willing to bet that the same reasons render a horse that finishes a track in either 50 or 60 seconds (and never in, say 55 seconds or 51.62 seconds) utterly implausible. As neither of us has any experience of a world where racing times are distributed binary there is no point in arguing what joint probabilities might be more reasonable to assume. $\endgroup$
    – loopy walt
    Commented Dec 15, 2021 at 17:03
  • 4
    $\begingroup$ Exactly, there is no reason to prefer any scenario over the other - all are possible, and regardless of what (non-zero) probabilities you put on them, A is overall the most likely to win. The only way that A is not uniquely the most likely to win is if you are 100% sure that either Scenario 1 or Scenario 2 are the case, and as you point out, there is no grounds to assume that. If you admit the possibility of all of these scenarios, A is most likely to win. $\endgroup$
    – Nuclear Hoagie
    Commented Dec 15, 2021 at 17:48
  • 4
    $\begingroup$ @NuclearHoagie Nope, you cannot average over scenarios. Only one of them can be correct, we just do not know which. $\endgroup$
    – loopy walt
    Commented Dec 15, 2021 at 18:30