Math guys??: Mathematics of streaks

This article suggests that given a 50/50 events (ie coin flips) there are event sequences where the expected odds are much less than 50%; Thus 50% outcomes in certain sequences does indeed imply a "hot hand"? Do this have implications in gamblingl(ie roulette)?



https://phys.org/news/2017-03-momentum-isnt-magic-vindicating-hot.html

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Suppose a researcher looks at the data from a sequence of 100 coin flips, collects all the flips for which the previous three flips are heads and inspects one of these flips. To visualize this, imagine the researcher taking these collected flips, putting them in a bucket and choosing one at random. The chance the chosen flip is a heads – equal to the percentage of heads in the bucket – we claim is less than 50 percent.

To see this, let's say the researcher happens to choose flip 42 from the bucket. Now it's true that if the researcher were to inspect flip 42 before examining the sequence, then the chance of it being heads would be exactly 50/50, as we intuitively expect. But the researcher looked at the sequence first, and collected flip 42 because it was one of the flips for which the previous three flips were heads. Why does this make it more likely that flip 42 would be tails rather than a heads?
Momentum isn't magic – vindicating the hot hand with the mathematics of streaks
Why tails is more likely when choosing a flip from the bucket. Credit: Miller and Sanjurjo, CC BY-ND

If flip 42 were heads, then flips 39, 40, 41 and 42 would be HHHH. This would mean that flip 43 would also follow three heads, and the researcher could have chosen flip 43 rather than flip 42 (but didn't). If flip 42 were tails, then flips 39 through 42 would be HHHT, and the researcher would be restricted from choosing flip 43 (or 44, or 45). This implies that in the world in which flip 42 is tails (HHHT) flip 42 is more likely to be chosen as there are (on average) fewer eligible flips in the sequence from which to choose than in the world in which flip 42 is heads (HHHH).

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Quote: bazooooka

This article suggests that given a 50/50 events (ie coin flips) there are event sequences where the expected odds are much less than 50%; Thus 50% outcomes in certain sequences does indeed imply a "hot hand"? Do this have implications in gamblingl(ie roulette)?



https://phys.org/news/2017-03-momentum-isnt-magic-vindicating-hot.html

>>
Suppose a researcher looks at the data from a sequence of 100 coin flips, collects all the flips for which the previous three flips are heads and inspects one of these flips. To visualize this, imagine the researcher taking these collected flips, putting them in a bucket and choosing one at random. The chance the chosen flip is a heads – equal to the percentage of heads in the bucket – we claim is less than 50 percent.

To see this, let's say the researcher happens to choose flip 42 from the bucket. Now it's true that if the researcher were to inspect flip 42 before examining the sequence, then the chance of it being heads would be exactly 50/50, as we intuitively expect. But the researcher looked at the sequence first, and collected flip 42 because it was one of the flips for which the previous three flips were heads. Why does this make it more likely that flip 42 would be tails rather than a heads?
Momentum isn't magic – vindicating the hot hand with the mathematics of streaks
Why tails is more likely when choosing a flip from the bucket. Credit: Miller and Sanjurjo, CC BY-ND

If flip 42 were heads, then flips 39, 40, 41 and 42 would be HHHH. This would mean that flip 43 would also follow three heads, and the researcher could have chosen flip 43 rather than flip 42 (but didn't). If flip 42 were tails, then flips 39 through 42 would be HHHT, and the researcher would be restricted from choosing flip 43 (or 44, or 45). This implies that in the world in which flip 42 is tails (HHHT) flip 42 is more likely to be chosen as there are (on average) fewer eligible flips in the sequence from which to choose than in the world in which flip 42 is heads (HHHH).

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Some incorrect conclusions reached from reading badly written explanations.
If the collected flips went in the bucket as described, then, on average, half of them would be heads. That's all you need to know.

Quote:

we claim is less than 50 percent

is an unsubstantaited claim. For all the maths and narative, that is not established at all. It is simply wrapped up in smoke and mirrors.

Hmm, but Imagine if you do 100 coin tosses and observe 50 “heads” and 50 “tails.” No problem.

But if you now observe the recorded sequence and begin to count backwards from 50 every time you see a “heads,”: you’ll always know how many “heads” remain in the sequence.

Necessarily, the number goes down by 1 every time you see a “heads” in the sequence.

And the "tails" number does not go down — it stays the same — every time you see a “tails” in the sequence.

Thus the probability, in this "restricted" population/sequence of flips, is such that the “next” flip in the sequence will be a “heads” is always lower if the “previous” flip was a “heads” than if it was a “tails.”


Here's a chart that restricts it to 4 flips and show the breakdown of different outcomes based on already observing a specific outcome (i.e. a 1 or more heads on the previous coin flip(s):

https://pixel.nymag.com/imgs/daily/science/2016/08/12/12-coin-flip-probabilities.nocrop.w710.h2147483647.gif



Anyhow; I have some bright friends who similarly track red vs black over 1000s of spins. They then play the opposite of whatever color has shown up disproportionately.

**I'm told over 10,000 spins the red v black population (at least based on internal casino data) always equates to very close to 50/50 (disregard the greens). But maybe over 2000 spins they can sometimes find a 47/53 type break down. These guys then pounce and hang in there for the 10000 or more spin "long run".

Hopefully these "mathematicians" have lost all their money playing roulette. At least, they deserve it for doing such math trickery.

Quote: RS

Hopefully these "mathematicians" have lost all their money playing roulette. At least, they deserve it for doing such math trickery.

Is it 'math trickery' to provide hope? Or is it more akin to pouring a drink for someone who could use it? I don't think people really distinguish between history and future spins, they always think they are 'in the present' and the present is some sort of "streak".

Everyone knows that if a dozen women come up and give you a kiss, the thirteenth may well be the one to use her knee instead. So enjoy the 'trend' as long as you can but realize full well that each and every spin is independent and that independent means its also independent of your hopes and expectations.

It's on the Internet so it must be true. (And, I'm a French model.)

Quote: QFIT

It's on the Internet so it must be true. (And, I'm a French model.)

Quote: bazooooka

Anyhow; I have some bright friends who similarly track red vs black over 1000s of spins. They then play the opposite of whatever color has shown up disproportionately.

**I'm told over 10,000 spins the red v black population (at least based on internal casino data) always equates to very close to 50/50 (disregard the greens). But maybe over 2000 spins they can sometimes find a 47/53 type break down. These guys then pounce and hang in there for the 10000 or more spin "long run".

I'm told your bright friends are suckers.
[Edit}
No I'm sorry. 'Suckers' is not a strong enough word.
'Stupid, ignorant idiots, destined to lose' might just about do it justice.

Quote: bazooooka

Here's a chart that restricts it to 4 flips and show the breakdown of different outcomes based on already observing a specific outcome (i.e. a 1 or more heads on the previous coin flip(s):

https://pixel.nymag.com/imgs/daily/science/2016/08/12/12-coin-flip-probabilities.nocrop.w710.h2147483647.gif

Not many here would be daft enough to waste there time reading flawed proofs of the gambler's fallacy.
I took a quick look and as soon as I saw that someone was taking the unweighted average of a set of ratios, I concluded I was reading the work of idiots and saved myself any more wasted time.
Well, just a tiny bit.
https://sinews.siam.org/Details-Page/hot-hands-streaks-and-coin-flips-how-the-new-york-times-got-it-wrong
.405 is the answer to a maths question: It is not the probability of any fair coin toss.

A coin and a roulette ball have no memory so future results cannot be predicted using past results. Given a large enough sample size of data anyone can find some sort of pattern. The problem is that the pattern is not repeatable with any certainty. Given any random situation, just because an event has happened in the past with regularity does not mean it will happen to a lesser extent in the future to balance out. It sounds logical but is completely false and is known as the Gamblers Fallacy.