# A coincidence isn't that unlikely?

#### blubblub

##### Member
A few days ago I had a brain thought. Often you hear people say if something unlikely occurs 'what are the chances'. Or in case something really unlikely occurs that it is almost a miracle. I'll use the word 'miracle' here just for a lack of a better word (I'm not a native speaker), not so much in the religious way.
That got me thinking about the actual chances. Not precise but in general. The chance of one very unlikely event is by definition then very unlikely. For example flipping a coin and it lands on its edge rather than head or tail. However, there are also many of these events that we could consider as near impossible. So, although getting a specific 'miracle' is unlikely, getting at least one is not so unlikely.
If we say that a 'miracle' is something that happens only with a chance of 1 in 10^9, but also concede that we can think of 10^9 possible events that would classify as a 'miracle', using the binomial distribution we would still have a 26% chance of at least one miracle happening.

Is my reasoning flawed somewhere, and if not is this a 'thing' as in does this have a name or something?

#### Karabiner

##### TS Contributor
You are correct. People usually ask "what are the chances" after the occurence of an event.
They neglect the sample space of millions of rare events that could potentially have happened.
Of course, the chance for any particular event is rare, but the chance that at least of these events
will happen is high.

If we learned that Mr. X won the statewide lottery, would the headline say "wow, what a miracle,
some Mr. X won the lottery, this is a one in ten million event, nearly impossible!" ?

#### blubblub

##### Member
Thanks for the fast reply. Glad that my reasoning wasn't flawed, since I know how tricky probability can become

Is there a name for this kind of fallacy(?) where you forget to take into consideration the sample space?

#### AngleWyrm

##### Active Member
If we say that a 'miracle' is something that happens only with a chance of 1 in 10^9, but also concede that we can think of 10^9 possible events that would classify as a 'miracle', using the binomial distribution we would still have a 26% chance of at least one miracle happening.

Is my reasoning flawed somewhere, and if not is this a 'thing' as in does this have a name or something?
The Gambler's Fallacy is a trick of memory, where the observer considers what has been observed but doesn't remember what hasn't taken place.

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#### Dason

Oh, I thought that it was about not having understood the idea of statistical independence.
That's mainly what is referred to when talking about gamblers fallacy. Gamblers have lots of break down in reasoning though.

#### AngleWyrm

##### Active Member
Gambler's Fallacy
What Is the Gambler's Fallacy?
The gambler's fallacy, also known as the Monte Carlo fallacy, occurs when an individual erroneously believes that a certain random event is less likely or more likely to happen based on the outcome of a previous event or series of events. This line of thinking is incorrect, since past events do not change the probability that certain events will occur in the future.

KEY TAKEAWAYS
• Gambler's fallacy refers to the erroneous thinking that a certain event is more or less likely, given a previous series of events.
• It is also named Monte Carlo fallacy, after a casino in Las Vegas where it was observed in 1913.
• The gambler's fallacy line of thinking is incorrect because each event should be considered independent and its results have no bearing on past or present occurrences.
• Investors often commit gambler's fallacy when they believe that a stock will lose or gain value after a series of trading sessions with the exact opposite movement.

#### Karabiner

##### TS Contributor
So the phenomenon the OP discusses (assigning probabilities to events which already have
happened) is not gambler's fallacy (which is about assigning probabilities to future events).

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Miracles in this content could equate to 'rare' event as @Karabiner mentioned. Could also be called anomaly or in engineering, values far from the mean are sigma six or greater events. Sigma being standard deviation.

I have always likened the gambler's fallacy to independence like @GretaGarbo mentioned. In particular, each turn on a roulette wheel or coin toss - given you can't know of of the physical properties and settings. Moreover - the idea that the next spin will be red since the last six have been black. A fair wheel doesn't care about the past spins, each new one is independent of the past - given it is a fair wheel or coin.

Rare events are possible, but have a low probability. However, as mentioned - if you are looking for any rare event in any domain - your probability of finding one increases just because the sampling spaces are increasing, but if you were to take how rare each was into account along with the ones you didn't see - they would still be rare.

To add to this - I will mention the 'sharp shooter fallacy'. Usually shooting at a barn then drawing the bullseye around the best cluster of shots after the final shot.