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Many people feel as if significant events are less probable. For example, some may feel as if the sequence of all heads on a coin is less probable than any other sequence. Or that the next lottery draw numbers being the same as the last winning draw is less probable. Of course, we know that each of those things are just as probable. It is no more likely for a sequence of HHHHH to occur than HTHHT in a fair coin for example.

But suppose a person does think that a sequence of all heads is more improbable. This, in wider society, is taken to characterize a failure in intuition. It is then used as an example of human beings not being able to understand probability correctly.

But after thinking about it further, I can’t see how those intuitions are wrong (or right for that matter). For starters, classes or categories of events don’t actually exist inherently. An event can be put into a class only by us. Only after we put an event into a category can we define a probability for that category.

If one puts the event of HHHHH into the class “sequence containing all heads and tails” and all other sequences into the class “sequence not containing only heads or tails”, then the HHHHH event is less probable. The HHHHH event is only equally as probable as the HTHHT event if one puts each coin toss sequence into its own respective class.

But there is no intrinsic apriori reason to put an event in one class over another. So when humans consider certain significant events more improbable, is it really a failure in intuition as many psychologists and some philosophers think, or is it that there is no correct probability of any event in the first place?

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  • I don't understand what you mean by "class" here. It feels like you just threw this term into the discussion, without first defining exactly what it means. Commented Sep 3, 2023 at 0:24
  • A class is what it normally means: a set or category of things. One can put an event into many different categories
    – user62907
    Commented Sep 3, 2023 at 0:43
  • Then I don't understand the sentence If one puts the event of HHHHH into the class “sequence containing all heads and tails” and all other sequences into the class “sequence not containing only heads or tails”, then the HHHHH event is less probable. Why does simply belonging to a class make it less probable? Commented Sep 3, 2023 at 0:46
  • Because events don’t have inherent probabilities. Classes do. And depending on which class you put an event into in your head, you get different probabilities.
    – user62907
    Commented Sep 3, 2023 at 0:48
  • Because events don’t have inherent probabilities That's quite an assertion. Proof? Commented Sep 3, 2023 at 0:50

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As far as I can tell, you are saying that some people mistakenly think HHHHHH is less likely than THHHHH because we wrongly equate the latter into 5 heads and a tail, which is more likely.

You might want to look into the psychology of human reasoning, and psychologism, and wonder what we can infer from common mistakes in reasoning.

Even if people usually make the same mistakes, I don't think we can infer that they are not mistakes. An argument ad populum might claim otherwise

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"If one puts the event of HHHHH into the class “sequence containing all heads and tails” and all other sequences into the class “sequence not containing only heads or tails”, then the HHHHH event is less probable. The HHHHH event is only equally as probable as the HTHHT event if one puts each coin toss sequence into its own respective class."

The error here is to conflate the probabilities of a sequence of five coin tosses with the probability of a single coin toss (the last one).

There is a further conflation here:

If one puts the event of HHHHH into the class “sequence containing all heads and tails” and all other sequences into the class “sequence not containing only heads or tails”, then the HHHHH event is less probable. The HHHHH event is only equally as probable as the HTHHT event if one puts each coin toss sequence into its own respective class.

If you consider random coin flips, then the sequence HHHHH is equally likely as "HTHHT", but that is because you have picked one member of the class “sequence not containing only heads or tails”, if you include all members of that class, then HHHHH is less likely than observing a member of the class, rather than specifically HTHHT.

The problem here is trying to take an intuitive approach to reasoning about probability, rather than actually performing the calculations (or a simulation) and getting bogged down with the ambiguity of natural language. The problem isn't with probability, the problem is in the woolly thinking in the question.

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    @Larsan saying that "probability is dependent upon sample space" is basically saying "the probability depends on what it is you are computing the probability of", which is a statement of the extreme obvious. This doesn't make it subjective, the choice of question you are interested in may be subjective, but once the question is stated unambiguously, the answer objectively follows. Commented Apr 30 at 10:17
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    @DikranMarsupial I would go at far as to say, the entire confusion in the original question can be boiled down to not knowing what question they actually want to ask
    – TKoL
    Commented Apr 30 at 10:21
  • @TKoL indeed, and perhaps an agenda in this particular case, as this user has asked many similar questions using a variety of different user names. Commented Apr 30 at 10:24
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    Taking shortcuts with math rarely works out well. Especially if probability is involved.
    – Scott Rowe
    Commented Apr 30 at 10:39
  • @Larsan ask an actuary, they can give you the probability that you ask for (note the more information about your circumstances, the more they can refine the answer). "OP picked a bad example" ;o) BTW the "sample space" for your question is "will die tomorrow" or "won't die tomorrow", unless you happen to be Schrodinger's cat, of course. Commented Apr 30 at 10:56

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