The “gambler’s fallacy”

[reddog]

Can some one explain some thing to me?

 

If I toss tails 9 times in a row, then tossing tails for the 10th time is still considered 50/50 as a single event (as per the article)

 

However, the odds of tossing a coin tails 10 times in a row must be 1000s to 1.

 

Therefore isn't it actually more likely that 10 times tails won't come up, and therefore heads is the more likely event for the 10th toss?

 

I know the answer must have to do with single events vs multiple events, but still can't quite explain it to myself.

Edited December 27, 2020 by reddog

[Wayward]

 

i understand the theory.  In terms of a coin flip - it's a 50/50 chance of heads or tails, and the next coin flip is also 50/50 - because the two events arn't linked.

But here's what i don't understand - if you flip a coin 1000 - you'd be more likely to see a mix of heads and tails, rather then all heads - so how does that fit in with the theory? 

Think of it this way...the coin doesn't have a memory. Simple.  It can be a thousand heads but on the 1001st spin that is unknown and irrelevant. 

[goldbug9999]

By the time you have thrown the 9th tail you have already achieved the one-in-thousands chance of that permutation, and the remaining throw is a 50/50 (same as all the previous ones).

Or ... imagine that you throw 10 coins all at once - the likelihood of a tail on any specific coin is 50/50. Throwing a coin 10 times vs 10 at once makes no difference probability wise.

This is really essence of the fallacy: that doing something random x times in a row is somehow different to doing x random things in parallel when it isnt. 

Edited December 27, 2020 by goldbug9999

[This time]

There are 1024 different permutations when tossing a coin ten times.

HHHHHHHHHH is one HHHHHHHHHT is another - they're therefore equally likely from the outset. The reason this throws people so much is that throwing 9 heads and a tail is more likely than throwing all heads if we don't care about the order as there are ten ways of achieving that result.

[Gigantic Purple Slug]

In the real world you should use previous history in informing the outcome of the next event, in the respect that previous history should indicate whether the event is truly priced correctly or not.

For example if someone tossed a coin 50 times and it came up heads each time, if the event was truly 50:50 then the odds of tossing heads 50 times in a row is so astronomically small that you would have good reason to believe the event was not actually truly 50:50 and was in fact biased towards heads. This should inform your future decision.

This method of thinking is more useful in the real world than applied to theoretical coin tosses (where pricing is exact or very close to being so), where determining probability of events is often difficult. However, if you can spot events are mispriced, so can other people. The trick is to spot mispricing on justifiable data before your counterparty lowers the return.

If you can clearly and unambiguously establish the probabilities on different outcomes in a game the return on your money will tell you that you shouldn't be playing it in the first place.

 

[A.steve]

 

Can some one explain some thing to me?

If I toss tails 9 times in a row, then tossing tails for the 10th time is still considered 50/50 as a single event (as per the article)

However, the odds of tossing a coin tails 10 times in a row must be 1000s to 1.

 

There are two perspectives that can help you get a handle on this.

Perspective 1: "It is a fair coin"

Coin-tossing (when spoken about by a mathematician) always requires the coin to be exactly fair - and the probabilities of heads and tails are exactly equal.  In the real world, this isn't the case.  There is some tiny possibility that any coin toss could land on its edge - and empirical studies have shown slight biases in real coins.  If I were to use slight of hand to switch-in a double-headed (or double-tailed) coin, of course, your probabilistic expectations will definitely be misguided.

One needs to consider cynicism... is it unthinkable that someone has duped you by using a known biased coin?  If not unthinkable, is it more or less likely than 1 in 512 (i.e. 1 in 2 to the power 9) that you've been deceived in the context in which you're making a judgement?

Perspective 2: "Where's the time-lord?"

If the coin is fair, then every coin toss has - by definition - a 1 in 2 chance of being heads and a 1 in 2 chance of being tails.   No further analysis is required - this is axiomatic.

The question about whether you will get 10 tails in a row depends greatly upon whether or not you already know that the past 9 coin-tosses count and if they all happened to be tails.  If you're looking for 10-tails and you know each coin-toss is 50:50 - then you're in a much better position after 9-tails than after a single tails or a heads.

If you can monkey around with time, you can create illusions.  This maths answer deals with the expected number of heads in a sequence of (fair) coin-tosses.  This concept was exploited by a stage magician a few years ago to show an implausible run of a single outcome by flipping a coin many thousands of times.  (It might have been David Blane - but don't quote me on that.)  He filmed himself flipping a coin - and had two advantages:  (1) He could get one 'free' result by only deciding if he wanted heads-or-tails after he'd had a run of either.  (2) He could dispose of the footage of all the sequences that did not meet the requirements of the illusion.  Of course, he could have done it even easier if he could individually drop the results he didn't want... he claimed he did not... it would have required some advanced video manipulation.

Summary

The probability of tossing 10 tails in a row - assuming a fair result of each coin-toss and no shenanigans - is 1 in 1024 ... i.e. 1 in 2-to-the-power 10.  It is unlikely but not implausibly so. Introduce parties who are motivated to deceive you... and you could easily be blind-sided by misapplication of an overly simplified model.

Quirky Related Annecdote

Consider the Monty Hall Problem.  I've always disliked it.  Arrogant, mildly-educated, mathematicians will tell you that the answer is that the contestant should always switch... but this is just their cognitive bias.  These people have a blind-spot to the possibility that the contestant may be treated specially by the gameshow host... who might hold additional information that was not available to them as a contestant.  If one includes suspicion that the contestant might have been told the 'rules' in a misleading way into one's model... then the correct choice becomes far more subtle.  In real life, it matters a lot which assumptions you make before you crank the arithmetic of the model you adopt.  Jules Verne demonstrated in his fiction that numbers are a great convincer...  mainstream media, and politicians, have taken this idea and run with it.  Some people feel that presentation of unsolved simple arithmetic problems, today, are more likely to be a misdirection than an opportunity to gain an advantage by calculation.  If one presents a subject with a puzzle they solve - then they will be emotionally invested in the outcome of their calculation... despite it being under someone else's control.  The only difficult bit is deciding what sort of calculation will fall within the competency of the "mark"...

 

Edited December 27, 2020 by A.steve

[winkie]

 

No difference in theory but (and its the same with MOTs) they are aware of what the national average pass rate is and know they might get a visit from an inspector if they differ significantly from that.

Exactly....there are other forces and leanings that see an outcome fits into the norm.

[erat_forte]

Test results, both at school and at the MOT centre, definitely vary a fair bit over the course of the day in a statistically predictable way.