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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?

Mark Andrews
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4 Answers4

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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.

Marco Ocram
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  • So which one should be looked at then? –  Dec 31 '22 at 00:03
  • 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. – Marco Ocram Dec 31 '22 at 08:38
  • So which one matters when assessing whether something is designed? –  Dec 31 '22 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. – Marco Ocram Dec 31 '22 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.

niels nielsen
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  • What counts as a previous trial? –  Dec 24 '22 at 01:27
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    @thinkingman, have a look at the wikipedia page for "Bayesian statistics". – niels nielsen Dec 24 '22 at 03:50
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    @thinkingman have a look at Design of Experiments as well: Design of experiments - Wikipedia https://en.m.wikipedia.org/wiki/Design_of_experiments –  Dec 28 '22 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.

Atif
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  • But chance can create patterns. just not with large samples –  Dec 30 '22 at 06:16
  • @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 '22 at 06:36
  • 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 '22 at 06:40
  • @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 '22 at 06:49
  • 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 '22 at 13:11
  • @temp_rt 3 heads in a row do not happen by chance. Nothing happen by chance. This is a necessary outcome of universe being created by Creator. "Obviously not true" how? Why this particular coin had 3 heads in a row at this time? It must be something in its design. How else is one coin different from others? Its like asking how do the revolution that deposed its king and introduced democracy happened in France? Why it didn't happen in England? There must be something unique to France to have it happen there. Why is there life on earth only? – Atif Dec 30 '22 at 14:24
  • Where is the evidence that the 3 heads was intended by a Creator? –  Dec 30 '22 at 17:05
  • Because everything is intended by Creator. Creator of universe made everything that exist other than Himself. He, the first cause, is never created and is eternal. – Atif Dec 30 '22 at 17:09
  • Please do not sell personal belief as an argument, as it isn't one, much less a good one. – Philip Klöcking Dec 31 '22 at 17:34
  • It don't matter that its a personal belief. Its a belief of more than one quarter of world population, orders of magnitude more than belief in any other philosophy. Don't matter here still. What matter here is that it answer the question, in a logical way, fully backed up by data. Infact anything else would be wrong (such as presenting things one is not sure of himself, having home-made philosophy nobody else know or care about or believe, have theory not backed up by data etc.) – Atif Dec 31 '22 at 20:13
  • If most of the world believed in goblins, would that mean goblins exist? Secondly, you have not provided evidence that a Creator is intending everything. –  Dec 31 '22 at 20:53
  • @thinkingman 1. No, it wouldn't. Number of people believing something don't make the argument any more or any less true than it is. 2. I mentioned number of people in a particular belief to counter the "Its your personal belief" as in "private, home grown one". 3. A thing being someone's belief don't make it more true and don't make it less true therefore it shouldn't be brought up here. See "Please do not sell personal belief as an argument" above which I replied. Now, if you two look at whats presented by me and by nature and take off your anti-religion biasnes, just look at whats in front.. – Atif Jan 01 '23 at 03:35
  • ..of you you can see proves, many, many proves. Look in my answer its there. Look in natural world around you as well for independent observation and confirmation of what I said. Here you go "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". There. A proof of existence of Creator. What is a pattern? Its repeated arrangement. Cannot happen by chance. What happen by chance cannot have pattern. Nothing around us happen in a pattern on its own... – Atif Jan 01 '23 at 03:41
  • ...How do diamonds form? Directed energy, directed pressure. Why directed? Because of some reason, some cause. That cause by same logic need other causes till the first cause. Look it up. You see milk in a shop being sold in boxes. You cannot pick one up arguing that since all materials exist in natural world they somehow on their own get themselves packed so there is no human contribution why pay then? Your argument will not fly because nobody ever observed things happening on their own. Continue this argument and you will end up in either jail or in mental asylum. – Atif Jan 01 '23 at 03:48
  • Faith is defined by saying that it is a belief that lacks evidence for support. In philosophy, Kant pointed out that it is our mind that likes to see reason and telos everywhere, over 200 years ago. Therefore, again: faith is not a valid argument for anything and lacks proof/evidence by definition. – Philip Klöcking Jan 01 '23 at 09:11
  • @Philip Wrong. Thats not what faith is. Faith is what one believe in dont matter that what one believe in is true or false. A faith in cookie monster is as much a faith as in things of equal mass following at same time. Kant didnt say that faith by definition is without proof either. I didnt gave faith as evidence of anything. For evidence I gave data. Only data can be evidence. – Atif Jan 01 '23 at 09:21
  • "If taking something to be true is only subjectively sufficient and is at the same time held to be objectively insufficient, then it is called believing [glauben]. (CPR B850)" You can use your own definitions and beliefs to your liking but it will not change facts. – Philip Klöcking Jan 01 '23 at 10:24
  • @Philip What fact? That someone somewhere said something? What about many, many others saying some other things at many, many other places? If you want definition of a word look in a dictionary, duh. Faith is defined as something considered to be true. Nothing more, nothing less. It may or may not be backed by facts. You are implying that each and everything anybody ever believe in is objectively false. Lets have a thought experiment. Suppose I dont know that proxima centauri is infact 3 stars system. I believe (wrongly) that its 2 stars system. Then I change my belief to it being... – Atif Jan 01 '23 at 10:33
  • ...3 star system. Do the star system now stop being a 3 stars system? Do my change in the belief change what it is? Ridiculous. – Atif Jan 01 '23 at 10:34
  • The normal terminology in philosophy differentiates between believing that something is true as the general category and opinion, faith, and knowledge as subcategories. The quoted passage of Kant is pretty much the standard view: knowledge is considered subjectively and objectively sufficient, faith is a belief that is held as subjectively indubitable but without sufficient objective evidence, opinion is both subjectively und objectively shaky. All of them are kinds of belief in the sense of "propositions that are believed". Also, faith is neither objectively true nor false, just unempirical. – Philip Klöcking Jan 01 '23 at 11:23
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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.

gnasher729
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