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You lose at cards. Suppose the chances that the other person was card counting, P, is 1/1000.

You lose at cards again. This time, the chances that this person was counting cards, B, is 1/100.

The chances of B and not P is let's suppose 1/4, as it involves very similar pathways, motives, skills, character etc.

What is the updated chance of P?

Is it a fallacy to not update your belief in P (the chances that they were originally card counting) assuming that card counting B would be linked to card counting P?

I think it's the fallacy of exclusion.

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The chances of B and not P is let's suppose 1/4

As written, this is contradictory with the other premises. I think you mean to say, the chance of not-P given B is 1/4. This also means the chance of P given B is 3/4.

So P(B) = 1/100, P(P|B) = 3/4 = P(B & P) / P(B)

P(B & P) = 3/400

Since P(P) >= P(B & P), this tells us that P(P) >= 3/400. So we can no longer assign a probability of 1/1000 to P; P must be at least 7.5 times more likely than that. There is not enough information to assign an exact value, however. To do that you'd need to formulate the problem in a way that permits a Bayesian analysis.

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  • that seems about right, thanks
    – user66760
    Commented Jul 22, 2023 at 4:10
  • why should we use BT?
    – user66760
    Commented Jul 22, 2023 at 4:35
  • Experientially, people probably could not observe the difference between 3:400 and 1:1000, unless they were willing to try a lot of times, which defeats the purpose of avoiding a rigged game. We want to avoid unwanted outcomes before losing, not learn from them. We want to be safe, not part of the fodder of millions of years of evolutionary failures. But probability is little help for that.
    – Scott Rowe
    Commented Jul 22, 2023 at 13:39
  • 1:1000 feels very different to 1:100 @ScottRowe to me anyway.
    – user66760
    Commented Jul 22, 2023 at 17:24
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    @doot_s If you had a thousand sided die, with a mark on only one side, and similarly a hundred sided die, how many times would you have to roll them both to get a sense that they were different? Some smarter person than I can probably say.
    – Scott Rowe
    Commented Jul 22, 2023 at 19:14

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