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Is it valid to consider a rare event more likely to have been produced by chance if previous trials exist?

Suppose John tosses a coin and thinks his mind can cause it to land on heads. He tosses the coin and it lands on heads 10 straight times. Suppose, for the sake of this example, that the coin couldn’t have been rigged, or that its probability is extremely low, say 1 in 10^50.

I am now tasked with figuring out whether his mind affected the result or whether it was chance. Personally, I would not believe it. The reason I wouldn’t is because I sort of imagine previous trials. I imagine that in human history, people must have tossed millions of coins. It is not surprising that eventually, at least one person would land a coin on heads 10 times. Note that in this case this isn’t exactly analogous to the inverse gambler’s fallacy. The inverse gambler’s fallacy suggests that previous trials must have occurred because a rare event occurred. But this isn’t what I’m suggesting.

I am not proposing that previous trials must have existed BECAUSE this rare event occurred. I am using my reasonable knowledge of previous trials having occurred to infer instead that this rare event shouldn’t be that surprising.

Is this reasoning sound? And what here can be considered a previous trial in the first place? For example, imagine I created a new kind of coin. This coin had special attributes and was a type of coin that never existed before, but yet still, could only land on heads or tails. If John now landed this coin on heads 10 times, my intuition/mind may not consider previous trials of coins (which are of a different type) as relevant.

This type of coin in my head, after all, had not been tossed before. What if, instead of a coin, it was a different object, and yet produced one of two results with a 50% chance, and John instead “tossed” that object? Does this make a difference?

My general question is twofold if any of this sounds confusing:

a) should previous trials affect whether or not one thinks the current trial is produced by chance?

b) what constitutes as previous trials? How “similar” must the previous trials be as an event to the current trial to be counted? What should this similarity be based on?

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4 Answers 4

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As is suggested by your other questions on the same topic, you are confusing two issues. One is the probability of one specific rare event happening, which we will call p. The other is the probability P of at least one rare event happening when a large number, n, of opportunities for them to happen are taken into account.

p is small but P tends to 1 as n gets very large.

The probability of John tossing ten heads is very small- about one in a million. The probability of at least one person tossing ten heads in basket of hundreds of millions of attempts is much closer to one.

The number of attempts you decide to include in your basket will determine the probability of at least one of them being successful. But the chance of one specific attempt being successful remains one in a million.

This principle is illustrated routinely in a number of ways. Consider,for example, the UK national lottery jackpot. The chance of you winning it in a week, assuming you bought a ticket, is less than a million to one. The chance of someone in your town winning is low but much greater. The chance of someone in your county winning is higher still. And the chance of someone anywhere in the UK winning in greater than 50-50. So the greater the number of attempts you include in your basket, the greater are the odds that at least one will succeed, although the chance of any one individual attempt being successful will remain low.

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  • So which one should be looked at then? Dec 31, 2022 at 0:03
  • You should look at whichever suits your current purpose! If you want to calculate your individual probability of tossing ten heads then use individual probabilities. If you want to calculate the probability of at least one person tossing ten heads at a world coin tossing event, use the basket technique. Dec 31, 2022 at 8:38
  • So which one matters when assessing whether something is designed? Dec 31, 2022 at 15:32
  • It depends on the context. There is no black and white answer. If you find a nicely written sonnet on a piece of paper you might assume it is not a product of random chance. If you find it in a vast wilderness littered with trillions of papers containing random text and populated by countless typing monkeys, you might consider it random. Dec 31, 2022 at 17:49
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Within the field of classical statistics, the key to understanding what is going on in your example is the concept of odds i.e., in a certain specified number of trials, what are the odds that you will toss ten heads in a row, in the course of one trial (where ten tosses constitutes one trial)? This excludes anecdotal evidence (all the other coin tosses ever performed in the history of coins, which were not part of your test trials). Odds get figured within the context of your trials; previous history has nothing to do with those odds.

Bayesian statistics allows for the inclusion of previous trials to figure odds.

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  • What counts as a previous trial? Dec 24, 2022 at 1:27
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    @thinkingman, have a look at the wikipedia page for "Bayesian statistics". Dec 24, 2022 at 3:50
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    @thinkingman have a look at Design of Experiments as well: Design of experiments - Wikipedia en.m.wikipedia.org/wiki/Design_of_experiments
    – user59124
    Dec 28, 2022 at 19:05
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Ofcourse it is. How else will you know?

You can do an experiment but then also you will be analyzing after the results have come. You can only look at what already happened.

If you see a pattern then you know that what happened didn't happen by chance. If you don't see any pattern then you can conclude randomness - which is another name for chance.

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  • But chance can create patterns. just not with large samples Dec 30, 2022 at 6:16
  • @thinkingman No sir, patterns by definition are there because of some mechanism. Probability of having patterns by pure chance is so low that universe would have end before any of it appear. This infact is what intelligence is, finding patterns. Things don't spontaneously arrange themselves in order, if it did then there would be no Second Law of Thermodynamics, entropy wouldn't only increase then, it would also decrease. Heck, entropy not even stay constant. Entropy is measure of disorderness in a system.
    – Atif
    Dec 30, 2022 at 6:36
  • Me tossing a coin and landing it on heads three straight times is a pattern, is it not? Does this mean it didn't happen by chance? Sure if it landed on heads 500 straight times, it would imply it is not by chance. But in both cases, it is still technically a pattern. It's just that in one case, the pattern comes from a lot of samples. There are cases of objects in nature that look like Jesus or spelling out Allah or a baby or a tree that mark out patterns. It doesn't mean they didn't happen by chance, does it? Dec 30, 2022 at 6:40
  • @thinkingman The answer to that is, you are using different theory for small amount of data than what you are using for big amount of data when the nature of data is same. Its like having a different theory of gravity at sub-atomic scale than for celestial bodies. Theories don't work that way. Your theory should cover all data of its kind, scale don't matter. Getting heads 3 times in a row is conceptually not different than having them 500 times in a row.
    – Atif
    Dec 30, 2022 at 6:49
  • Are you saying that getting heads 3 times in a row is not explainable by chance? Because saying that is obviously not true. Now if you admit that it can be done by chance, then you admit that chance can create a pattern. But that contradicts your earlier statement where you said chance can’t create a pattern Dec 30, 2022 at 13:11
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If an event has a probability say 0.5, then it will happen quite often. Happening 10 times in a row is quite unlikely, but it will happen. 20 times is very unlikely.

Sometimes many people will make observations. Unlikely events will happen to some people. Say everyone in China throws coins until they throw “tails”, once a day for a year. A sequence of 38 heads is quite likely to happen because we repeat the observation so often. So very unlikely events can happen.

But at some point you start thinking ehrtet that probability is really 0.5. Let’s say I told you that you have a fair coin. I’m a very honest person and very hard to deceive. You estimate the chances that I gave you a coin that always throws tails is about 1 in s million. So after 20 heads the chance of coincidence and the chance of me cheating are the same.

After 40 heads you give the coin to a famous physicist with lots of lab equipment and a famous stage magician, and they both examine the coin most carefully and assure you the coin is fair. But 100 heads in a row is so unlikely, you can assume that it is not a fair coin, whether you can prove it or not.

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