From the Wall Street Journal
http://www.wsj.com/articles/the-hot-...ead-1443465711
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The ‘Hot Hand’ Debate Gets Flipped on Its Head

The new paper, written by Joshua Miller and Adam Sanjurjo, begins with a riddle. Toss a coin four times. Write down what happened. Repeat that process one million times. What percentage of the flips after heads also came up heads?

The obvious answer is 50%. That answer is also wrong. The real answer is 40%—and the authors say their correction should alter years of thinking about the hot hand.


The fallacy of the hot hand was established in a classic 1985 study that has since become a part of the social-sciences canon. The paper’s conclusion—that the appearance of shooting streaks was a misreading of randomness—was so counterintuitive that many refused to believe it. The uproar hasn’t abated over the years, yet even the most promising follow-ups found only a tepid hand. The feeling that you can’t miss after making several shots in a row was still a “massive and widespread cognitive illusion,” as the Nobel Prize winner Daniel Kahneman has written.

Nobel laureates think about the hot hand because it’s a bias that shapes important decisions. For these academics, the hot hand isn’t an isolated basketball occurrence. It’s an accessible example of how human beings behave with consequences for almost every industry.

Now, though, comes the most intriguing argument that human intuition wasn’t wrong. A basketball player who shoots the same percentage after a streak of makes as he does after a streak of misses was long accepted as proof against the hot hand. Miller and Sanjurjo’s paper asserts it’s actually evidence of the opposite.

“People were right to believe the hot hand exists,” said Sanjurjo, an economist at the University of Alicante in Spain.

Their breakthrough is the surprising math of coin flips. Take the 14 equally likely sequences of heads and tails with at least one heads in the first three flips—HHHH, HTHH, HTTH, etc. Look at a sequence at random. Select any flip immediately after a heads, and you’ll see the bias: There is a 60% chance it will be tails in the average sequence.
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Notice this is not looking at conditional probabilities like,

P(2nd flip is heads GIVEN 1st flip is heads)

which must equal 50% by independence of the coin flips.

Rather, this is saying that if you flip a coin 4 times and IF there is at least one heads in the first 3 flips, THEN

P(randomly picked heads in the first 3 flips is followed by a heads) = 40%

Actually, it's 17/42 or about 40.48% if you do the calculation.

I suppose if statistics were taken in this way to disprove the "Hot Hand" theory then this observation could be important.

It matters how statistics are done.


PairTheBoard

Looks like a Monty Hall version.

And it's retrospective, not prospective like coming happenings are.

Title change suggestion: Coin flips only 40% likely to have been flipped heads after a heads.

I see it like they "waste" the times there has been more than one heads immediately before, when your criterion is to look only at the last heads.

In a 4 hand sequence like HHHH do they count the heads after heads appearing 1 time or 3?


HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT
THHH THHT THTH THTT TTHH TTHT TTTH TTTT

i see 12 vs 12 here in the 14 qualifying sequences. What are they talking about?

Also why 4 and not 16 or 128 sequences? I still do not see how the people need to do the statistics to arrive at 40%?

http://andrewgelman.com/2015/07/09/h...is-a-hot-hand/

You don't. Statistics were done with basketball players to prove that the "Hot Hand" theory was bogus. Those statistics found that a player shooting after making a shot was no more likely to make the next one than usual for him. The idea here is that if they took those statistics the way the coin flips are being done here then those basketball statistics actually proved the "Hot Hand" theory rather than disproved it.


PairTheBoard

Can we produce a closed form solution for this probability for the arbitrary case of a sequence of n coin flips? Does the probability go to 50% in the limit? Why is it clear that all the sequences with at least one head in the first n-1 flips are equally likely? Can anyone explain where the bias is coming from?


PairTheBoard

I think the WSJ reporters blew it in their summary of the finding:

The quantity described here is the conditional probability of heads after heads and is just 0.5 as intuition suggests.

Where you get into trouble is if you do the following:
1) Take a million fair coins.
2) Flip each four times
3) For each coin, compute the relative frequency of heads-after-heads in that sequence of four flips.
4) Compute an equally weighted average of those one million values from step 3.

If you do this, you might expect to get something close to 0.5, but you won't.

Right. And the WSJ description was at best vague. However, further down they give the book's description which is more clear. And I think if you put it in the terms I did (see below) you get the counter intuitive flavor of it.


This is the description from the book (see last WSJ paragraph from OP)
----------------
Their breakthrough is the surprising math of coin flips. Take the 14 equally likely sequences of heads and tails with at least one heads in the first three flips—HHHH, HTHH, HTTH, etc. Look at a sequence at random. Select any flip immediately after a heads, and you’ll see the bias: There is a 60% chance it will be tails in the average sequence.
------------------

And my OP description of the probability in question:

"this is saying that if you flip a coin 4 times and IF there is at least one heads in the first 3 flips, THEN

P(randomly picked heads in the first 3 flips is followed by a heads) = 40% "


PairTheBoard

Only thing you need to know is it's 0.5 probability for heads when you are about to flip, no matter what's happened before. Question that and you question all of reality. The coin doesn't have a memory. How is it NOT supposed to be 0.5?

All the abracadabras can kiss my ass. If and if and if, select this and that from the past, show some athlete concentrating on throwing it in. He is not the one to change the natural laws, doesn't matter how tough and impressive he looks.

Know I'm saving the already saved here, but just in case...

No, I don't think this is accurate. Here are the 14 sequences with at least one heads in the first 3 flops:

HHHH
HHHT
HHTH
HTHH
THHH
HHTT
HTHT
HTTH
THHT
THTH
TTHH
HTTT
THTT
TTHT

In those 14 sequences there are 24 heads in positions 1...3. Of those 24 heads, 12 are followed by a heads and 12 are followed by a tails. Exactly as you'd expect.

To get the puzzle, roughly speaking, you need to compute an average for each sequence and then an average of those averages.

Or maybe the "Two Child" (Boy or Girl) paradox
https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
------------------
Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Gardner initially gave the answers 1/2 and 1/3, respectively; but later acknowledged that the second question was ambiguous.[1] Its answer could be 1/2, depending on how you found out that one child was a boy. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[3] and Nickerson.[
-------------------------------

PairTheBoard

HHHH = 3/3 = 6/6
HHHT = 2/3 = 4/6
HHTH = 1/2 = 3/6
HTHH = 1/2 = 3/6
THHH = 2/2 = 6/6
HHTT = 1/2 = 3/6
HTHT = 0/2 = 0/6
HTTH = 0/1 = 0/6
THHT = 1/2 = 3/6
THTH = 0/1 = 0/6
TTHH = 1/1 = 6/6
HTTT = 0/1 = 0/6
THTT = 0/1 = 0/6
TTHT = 0/1 = 0/6

((6+4+3+3+6+3+3+6)/6)/14 = 34/84 = 40.4%

PairTheBoard -
------------------
And my OP description of the probability in question:

"this is saying that if you flip a coin 4 times and IF there is at least one heads in the first 3 flips, THEN

P(randomly picked heads in the first 3 flips is followed by a heads) = 40% "
-------------------------------


Yes but that's not P(randomly picked heads in the first 3 flips is followed by a heads) in a random sequence containing at least one head in the first 3 flips.

I think the advantage to my description is it states it as a straight forward probability which can be computed as Aaron W. shows above.

Here's another description.

Take a coin. Flip it 4 times and record the result. If the first 3 flips are tails flip it another 4 times. Repeat until you get a sequence with at least 1 head in the first 3 flips. The sequence may have 1, 2, or 3 heads in the first 3 flips. Randomly pick one of them. Then look at the following flip. The probability that flip is a heads is 17/42.

That's what I mean by

"this is saying that if you flip a coin 4 times and IF there is at least one heads in the first 3 flips, THEN

P(randomly picked heads in the first 3 flips is followed by a heads) = 40%" (approximately - as in the article)

Note: In the OP I gave the exact calculation of 17/42


PairTheBoard

Length 16 sequence



Length 64 sequence


So the lesson is to stop doing stupid things and consider the entire sequence.


Also to be fair i define as hot hand only something that is at least HH already and then we test it. Then i go to even HHH etc...


And you better believe it in real life, where you do not have fair coins and correlations exist, there is a hot hand real effect. Stocks have that also. You play different if you have the attitude of a winner than if you are trying to stop the madness or in between. Something is working when you have a streak sometimes, you have improved the chances to be successful even a bit more.

In poker you must have seen for example that the best players destroy the others with streaks when lucky (it builds on their luck). They tilt opponents to death if they get lucky in a sequence of all ins (even better if its bad beats) and then boom they explode and find themselves pushing all in at flops 100bb deep with only top pair after a reraise or pushing all in preflop with TT etc.

That's to be expected. As the length increases, the bias decreases. But even at length 64, the calculation leads to a nearly 1% bias.

It might be easier to see what's going on for the case n=3. Below are the 6 equally likely sequences with at least 1 head in the first two flips. The probability shown next to each is the likelihood a randomly chosen head from the heads in the first 2 flips is followed by a head. The probability at the bottom is the total probability for that event.

HHH 2/2
HHT 1/2
HTH 0/1
HTT 0/1
THH 1/1
THT 0/1
----------
1/6 (1 + 1/2 + 1) = 5/12 = 41.7% apx

So the bias actually increases from n=3 to n=4.

Notice the 2 heads followed by heads in HHH are give equal weight as the 1 head followed by a head in THH.


PairTheBoard

The only way this could debunk statistical evidence against say Kobe Bryant's "Hot Hand" would be if the statistics randomly selected 1 shot Kobe made in each game and looked at the following shot. It seems unlikely they would have done it that way. Almost certainly they would have just looked at every shot Kobe made all season and checked to see if he made the following shot. This would not be relevant to that statistical method. And as egj has pointed out, coin flips done that way would show 50%.

Of course both cases above restricted to non last shots.


PairTheBoard

This. These guys deserve a hot hand- and by that I mean a fireplace poker with the end shaped like a hand- shoved right up their asses.

Its a mistake in my opinion to try to look for eg basketball shots hot hands in any other way than shots made at the same time (same foul event) (i mean try to look there first before other things dont mix it up yet). Or after say 1-2 successful (2/2 both in) shots that took place (ie a guy has shot 4/4 or 6/6, see what comes next). Yes of course you can check in a game too overall at any point but if there is a bias it must draw from the current situation that just took you there. I would only look first shot and then second shot looking for success then success or failure then success and try to see something (not mix things from previous minutes ago shots, at least at first studies). Then look for the 4/4 or 6/6 cases too (that defines hot trend). Same with 3 pointers. I would only look after 2/2 three pointers say vs any miss before success cases. All those are not very clean of course because they depend on the flow of the game too that may force a team to play desperate for 3 pointers affecting the success rate. (So maybe do tests during periods that do not qualify as excessively stressful for a team. Then maybe also do tests when the team is under extreme stress to catch up - but i mean do not originally mix everything).


I would also look for this thing; A case where player failed in both free shots and now is called in a new one later. That matters too. He has to break the nasty trend and re-establish himself and that is a stress factor.

Real life testing of such things is far more complicated and can mix up all kinds of biases that may even counter each other. Ie its a stress to try to succeed after failures that are remarkably soul crashing but at the same time you are now more focused and less careless.

What i also want to study is this;

If in soccer a team equalized another team 10 min or less before the end of the match what is the probability they win the match also (adjusted for bias due to home field advantage ie do the reverse studies where possible). Also probably choose only close matches where both teams are strong, not matches where one is very strong and the other team scored near the end to equalize by luck say after the first team lost a ton of chances. Although analysis like that can happen too to draw other conclusions but it must take care of the bias that a stronger team always has anyway.


Try also this other test. Find games that one team is much stronger than the other and they have a hard time scoring and at some point after many efforts they open the score finally. I bet the chance to score another now rises massively. I mean the first one comes very hard but the next 1 or 2 even come a lot easier. The teams play very open after that as the defending team needs to score now and the attacking favorite to win team is suddenly free from their stress of not scoring and possibly losing points in the match they were supposed to win. The complete liberation from stress is a game changer it seems.

Returning to basketball; You may of course run tests in exhibition events too that are significantly cleaner from side factors.

I would run tests for top (very famous and supposed to win by definition) players in exhibition events where they are given as information that eg a woman or an amateur player performed 7/10 3 pointers or something good to see if they feel the stress to not come short. Or cases that a big rival, has 9/10 say vs not knowing anything. Then maybe test all kinds of sports for who tries second vs first. Eg in long jump the second guy knows what be has to beat etc. Tons of data to study there.

If you have to do a semi-bs thesis on these things at least study a ton of interesting cases once and for all.

Speaking of all kinds of tests see also true real life coin flips (without the throwing up first thing) that you "drop" (= adiabatically/slowly let go, eg very slowly moving the coin till it fails to stand and drops from some edge - then reverse starting face and repeat etc) from a known height from eg a desk from the same initial position as much as possible. Then plot the results as function of the rising height (say you have a way to gradually change the height (and measure it accurately) without affecting anything else in the set up and the ground geometry the coin strikes.

I bet a great deal of determinism emerges but there is still significant chaos overall.

Penn and Teller had a show some years back where they looked at a guy who claimed that if you spin pennies they end up tails more often than heads (or visa versa). P&T supposedly authenticated it. I tried it with both old and new type pennies and could not find the effect. I have to call B.S.


PairTheBoard

Craziness! I've always had a weird feeling about these sabermetric guys who try and boil everything in sports down to numbers. Does anyone here not believe in the hot hand in sports? If a guy is hitting a bunch of shots in basketball, you feed him the ball, right? If Big Pappi is crushing everything he sees, you walk him, right?

PTB,

The end of Contact, the novel...

It's probably possible that over really long series (think 10^xxx) something might turn up that h/t is actually 40%.

Does not mean it is, does mean it is depending on the length of the stretch.

Nobody here really knows anything about probability or super-long series, at least not as much as they would like to. Things can and may look fixed but just might be, cobber.

I don't have a problem with sabermetrics. At least, the ones who get it are able to accept that noise is a dominant factor in their analyses, and don't make those types of mistakes.

I believe the hot hand can be real. I don't believe that analyses so far correctly capture the concept. Because the operationalized definition is that any made basket (or whatever) increases the probability that the next basket will be made, it doesn't really match the experiential feel of having a hot hand.

There's clearly a psychological element in play, both in the offense and in the defense when a hot hand is identified. It's not like so-and-so just made a layup, so next time we should give him the ball because he's got a hot hand. We know that there's such a thing as "being in the zone."

People believe in lots of things and make strategic decisions based on those beliefs all the time. But that doesn't mean those beliefs are a reality. Nor does it mean that those beliefs aren't a reality.

Yeah, I'm not ****ting on sabermetrics. There have been a lot of useful contributions from it. The increased use of shifts in baseball is a great example.

What I'm wary of are those people who would argue the stats are it and nothing else. For example, there are plenty of those guys who will argue "the zone" is imaginary. All you're seeing are purely natural runs of no statistical significance, just like if you flipped a coin and happened to get 6 heads in a row. What I believe is that humans are very complex machines in constant need of calibration and fine tuning. When someone, say an 80% free throw shooter, gets locked "in the zone," he is really a 90% free throw shooter (or some number higher than 80) for that period. Later, after he's cooled off, he may lose his fine tuning and be at or below 80% for a period.

The key things to notice in

P("a Heads" is followed by a heads) = 17/42 = apx 40%

is how "a Heads" is selected and how in the method of selection the sequence HHHH counts no more toward the calculated probability than TTHH.


PairTheBoard