Is it valid to consider previous trials when deciding whether a rare event occurred by chance?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.