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Suppose I predict that the lottery numbers that will be won tomorrow are a particular sequence and it comes true. Is this event less likely than it occurring without my prediction?

Intuitively, it seems yes, but after thinking about it, I fail to see how the likelihood of those numbers arising by chance would depend in any sort of way on my mind.

Even if I were to predict the lottery numbers three straight times, it seems that those numbers arising by chance have equal likelihood.

Why then does the former seem more impressive? Is it because if they are predicted, it increases the likelihood of cheating or some other process?

thinkingman
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  • Good question and I see some high quality answers. Speaks to the complexity of the topic. My money is on the OP conflating two very different ideas in re chance. Sometimes knowing a little math goes a long way towards bringing some clarity to problems. – Agent Smith May 17 '23 at 05:34

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The difference is not prior probability, but salience. If S is the winning sequence and pred(S) is the event that the winning sequence was predicted, then P(S | pred(S)) = P(S); whether or not you predict it, the probability of that particular sequence doesn't change.

However, a winning sequence that you predicted is far more salient to you than a typical winning sequence. Salience is about how relevant an event is to your personal situation and goals. You put time and effort into your prediction, and the "payoff" of your action was wildly out of expectation with the time and effort put in. Even if you didn't bet, but merely observed the result, you were still trying to get the right prediction, so it's still a psychological "payoff" of sorts if you do happen to get it.

A similar, though slightly different, situation often comes up in science; if you predict a pattern and then test it and indeed the pattern is there, this is a far more significant finding and much better science than if you had simply observed the same pattern in your data, after the fact.

Let me explain. Suppose you are a scientist and you observe some data, and you notice that the data fits a certain pattern that is very unlikely based on your background assumptions. Wow, you say, the chances of that happening under our background assumptions are incredibly low! I'm really onto something, and we'll have to revise our background assumptions!

But that's fallacious reasoning. The problem is that there are a very large number of individually unlikely patterns the data might fit, and so the chance that the data fits any particular unlikely pattern (under the background assumptions) could actually be fairly high. So you have not demonstrated a need to revise the background assumptions.

It would be far more impressive if you had predicted beforehand that the (unlikely-under-background-assumptions) pattern would occur, and it did. Because then, there's only one pattern you're testing for, and it is indeed very unlikely it would occur. So if your prediction comes true then you have indeed made a good case for revising the background assumptions.

causative
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  • When you say that it would be far more impressive then, it is NOT because the observed pattern is now less likely, but rather that it is MORE likely that your background assumptions about how the pattern is coming out may need to be revised, correct? – thinkingman May 16 '23 at 21:11
  • @thinkingman yes. Impressive scientific papers are those that provide strong evidence that the background theory needs to be changed. – causative May 16 '23 at 21:13
  • Fair enough @causative. The more pressing question then becomes when and at what threshold of "impressiveness" must we reach before questioning the background theory. For something like psychics, this question seems to be unanswerable. What are your thoughts? – thinkingman May 16 '23 at 21:26