Do this have implications beyond basketball (ie roulette)?

http://nymag.com/scienceofus/2016/08/how-researchers-discovered-the-basketball-hot-hand.html

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What the table shows, convincingly, is that due to what had been an as-yet-undiscovered quirk of the math, there is a cold hand built into the very laws of probability. The probability of getting tails on any individual flip is, of course, always 50 percent. But when you have a finite number of coin flips — or shot attempts, or any other probability-based event — the sequences with consecutive identical outcomes can only be arranged in so many ways. As a result, a given flip of heads is more likely to be followed by tails than by another heads.

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The coin flip argument where the "cold" is built in of course is bullshit. What the chart shows is the odds of having two heads in a row which they claim is 40% in a trial of four. There are 16 outcomes of 4 flips:

Success / Opportunities

TTTT - 0 / 0

HTTT - 0 / 1

THTT - 0 / 1

TTHT - 0 / 1

TTTH - 0 / 0

HHTT - 1 / 2 Y

HTHT - 0 / 2

HTTH - 0 / 1

THHT - 1 / 2 Y

THTH - 0 / 1

TTHH - 1 / 1 Y

HHHT - 2 / 3 Y

HHTH - 1 / 2 Y

HTHH - 1 / 2 Y

THHH - 2 / 2 Y

HHHH - 3 / 3 Y

Total - 12 for 24 = 50% - the odds of tossing two hits in a row in a four shot trial is 50%.

So I don't know the math where they get .40 and the quote "As a result, a given flip of heads is more likely to be followed by tails than by another heads."

Help me out here.

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https://mindyourdecisions.com/blog/2015/10/06/the-hot-hand-is-real-understanding-a-controversial-statistics-paper-game-theory-tuesdays/

""Jack takes a coin from his pocket and decides that he will flip it 4 times in a row, writing down the outcome of each flip on a scrap of paper. After he is done flipping, he will look at the flips that immediately followed an outcome of heads, and compute the relative frequency of heads on those flips. Because the coin is fair, Jack of course expects this conditional relative frequency to be equal to the probability of flipping a heads: 0.5. Shockingly, Jack is wrong. If he were to sample 1 million fair coins and flip each coin 4 times, observing the conditional relative frequency for each coin, on average the relative frequency would be approximately 0.4.""

and https://www.wsj.com/articles/the-hot-hand-debate-gets-flipped-on-its-head-1443465711

Quote:bazooookaOnce you "restrict" situations to when a head follows a heads flip then it seems to come out closer to 40%. This restriction makes certain sequences less than 50% - even though each flip is indeed 50/50.

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https://mindyourdecisions.com/blog/2015/10/06/the-hot-hand-is-real-understanding-a-controversial-statistics-paper-game-theory-tuesdays/

""Jack takes a coin from his pocket and decides that he will flip it 4 times in a row, writing down the outcome of each flip on a scrap of paper. After he is done flipping, he will look at the flips that immediately followed an outcome of heads, and compute the relative frequency of heads on those flips. Because the coin is fair, Jack of course expects this conditional relative frequency to be equal to the probability of flipping a heads: 0.5. Shockingly, Jack is wrong. If he were to sample 1 million fair coins and flip each coin 4 times, observing the conditional relative frequency for each coin, on average the relative frequency would be approximately 0.4.""

and https://www.wsj.com/articles/the-hot-hand-debate-gets-flipped-on-its-head-1443465711

0- tails, 1- heads . # of flips directly after heads . # of heads after heads

0000 . 0 . 0

0001 . 0 . 0

0010 . 1 . 0

0011 . 1 . 1

0100 . 1 . 0

0101 . 1 . 0

0110 . 2 . 1

0111 . 2 . 2

1000 . 1 . 0

1001 . 1 . 0

1010 . 2 . 0

1011 . 2 . 1

1100 . 2 . 1

1101 . 2 . 1

1110 . 3 . 2

1111 . 3 . 3

Overall, there are 24 flips directly after a heads and 12 heads directly after heads. I bolded all flips directly after a heads (or 1).

Did I make a mistake? Given that the answer I get is 0.5 (12/24 = 0.5) and logically the answer should be 0.5, I don't think I did.

Whatever this website is you're reading now, you should stop reading it. They're using MustangSally-type math. No good.

"""The researchers began their paper with a basic example to illustrate their findings: an experiment in which a person throws a coin in the air four times. "We noted the results of each throw, and calculated the percentage of heads tossed immediately after another head for all possible throw sequences. The results were unexpected: the proportion is not 50%, as we intuitively believe, but actually around 40%," Sanjurjo tells us, adding that "these results suggest the existence of a bias."""

Quote:bazooookahttps://goo.gl/8qFrcW

"""The researchers began their paper with a basic example to illustrate their findings: an experiment in which a person throws a coin in the air four times. "We noted the results of each throw, and calculated the percentage of heads tossed immediately after another head for all possible throw sequences. The results were unexpected: the proportion is not 50%, as we intuitively believe, but actually around 40%," Sanjurjo tells us, adding that "these results suggest the existence of a bias."""

From the abstract:

"The magnitude of this novel form of selection bias generally decreases as the sequence gets longer"

AKA variance

Quote:bazooookahttps://goo.gl/8qFrcW

"""The researchers began their paper with a basic example to illustrate their findings: an experiment in which a person throws a coin in the air four times. "We noted the results of each throw, and calculated the percentage of heads tossed immediately after another head for all possible throw sequences. The results were unexpected: the proportion is not 50%, as we intuitively believe, but actually around 40%," Sanjurjo tells us, adding that "these results suggest the existence of a bias."""

That paragraph is just mathematically wrong. Absolutely wrong way to combine the subsets of results. You never take an unweighted average of ratios. To get the result, you sum the original numerators and sum the original denominators and then work out the ratio. And behold, the correctly calculated ratio is 0.5

https://twitter.com/jben0/status/657331384604823553/photo/1

http://wizardofvegas.com/forum/gambling/tables/29032-math-guys-mathematics-of-streaks/