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Suppose a random person comes up to you and says "Think of a number between 1 and 10." You think of one. He guesses it correctly. You seem slightly surprised but ask him to do it again. He does it again. He ends up doing this successfully six straight times. Let's call your degree of belief in him using his mind to successfully guess your number d. You are apriori an ardent skeptic and believe that psychic abilities are a scam but your d has now gone significantly up.

Now, suppose you learn that he had been doing this to people for a year. Every single day, he'd go up to ten people and try to guess what number the other person is thinking of. Your degree of belief d changes. In fact, it becomes very minimal if not nonexistent.

And yet, on that day, it wouldn't have mattered if he did this to 50 other people, 10,000 other people, or 10 million. The probability of him guessing your number 6 straight times was 1 in 10^6 either way.

Is it rational to change your degree of belief in something depending on how many opportunities of an event there were in the past? Many would say yes, but this introduces a new problem: what defines an opportunity? In this case, should an opportunity only consist of how many times that person guessed a number before? What about how many times that person made other sorts of predictions? What about how many times other people made predictions? What about how many times opportunities arose for a meaningful event to occur, such as finding a lost friend after a year? How do we differentiate these classes?

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    Because events in life are typically not statistically independent. Gambler's fallacy only applies when such independence is known to hold, e.g. because the mechanism of generating those events is known, as in gambling. That is not the case in your guessing scenario, where the guessing mechanism is unknown, and it is reasonable to doubt that the guesses are independent random events.
    – Conifold
    Jan 22 at 8:02
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    To me, this question devolves into "how do I make a good belief model?" Deciding which factors and which cases do influence your beliefs is just part of doing that. So there is no one true answer to this question.
    – Dave
    Feb 23 at 19:39
  • "One nuclear bomb can ruin your whole day."
    – Scott Rowe
    Jul 22 at 0:40

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Say there are 10 people and there are 100 numbers to choose from.

The first person chooses a number and I'm supposed to guess it. The chances that I'll get it right is 1/100 and say I did get it right, it was 78. Now suppose 78 is no longer among the list of numbers that can be chosen. The next person chooses another number. My chances of guessing that number correctly is higher, 1/99. The previous probability affects the next probability. This is not the gambler's fallacy (dependent probability)

Next version of this guessing game is such that the number chosen can be repeatedly selected. If so the probability of me guessing the correct number of the first person = the probability of me guessing the correct number of the second, third, ..., tenth person = 1/100. The subsequent probability is unaffected by the preceding probability. To think it is affected is the gambler's fallacy (independent probability).

To assume the person who has guessed the correct number of n other people is likely to get it right with you is a gambler's fallacy if you can choose any number, even the ones the n other people had chosen

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  • Still, the difference between 1:100 and 1:99 or even 1:90 is not really noticable. People couldn't distinguish the fallacy versus random situations, so it is understandable that they don't see the difference.
    – Scott Rowe
    Jul 22 at 0:44
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If I manage to guess several numbers in a row correctly then either I am very lucky or I am cheating. It’s up to you to decide what is more likely.

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