Math guys??: Mathematics of streaks
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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Some incorrect conclusions reached from reading badly written explanations.Quote: bazooookaThis 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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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.
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.Quote:we claim is less than 50 percent
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".
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".Quote: RSHopefully these "mathematicians" have lost all their money playing roulette. At least, they deserve it for doing such math trickery.
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.
Quote: QFITIt's on the Internet so it must be true. (And, I'm a French model.)
I'm told your bright friends are suckers.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".
[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: bazooookaHere'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.
Quote: davethebuilderA coin and a roulette ball have no memory so future results cannot be predicted . . .
True but roulette is not played solely with a little white ball and a roulette wheel, it is played with people and those people do have memory and hopes and dreams so asking the players to consider the little white ball rather than their own hopes and dreams is not likely to gain you any friends.
Note: I sure do like that "Casino Enemy No. 1" tag line.
I've fallen for such false reasoning and when you add in a festive atmosphere, low cut evening gowns, perfume, alcohol, music, the sounds of chips and coins, etc. do you really think those who are not already enemies of the casino will think clearly. Those annunciator boards showing recent results bring in alot of money. Half the players are followers of the board, half are contrarians, but few ignore it as being utterly unrelated to the next spin.
I had a physics teacher in high school that used a bit of trickery like this to explain that lights do not emit light and that in fact they are "dark suckers" similar to a black hole. The closer you get the more dark they suck in and the further away the less powerful they are, etc, etc. At the end of the day all of us idiotic high schoolers were like "omg mathematically he's right" then he promptly made fun of us for accepting a couple of his base lines where he did some trickery like this that were essentially false claims.
He also had another experiment, which I think even today is still cute, but it involves similar trickery. Basically, you can "mathematically prove" a 50/50 is ONLY a 50/50 if you believe it is. What it boils down to is sampling size. Say craps pass/don't pass was a perfect 50/50. If someone only played craps 10 times in their life for 1 throw only... Then their results will be PURE variance. It will NOT be 50/50 it will literally just come down to luck. It's the same idea with a coin toss or any 50/50 (or even other odds)... The idea is if the person doesn't get enough trials in their lifetime, then from the perspective of their lifetime the event is NOT a 50/50 (or whatever the true odds are). One could use math to show that over multiple lifetimes it does come out to the true odds, sure... BUT do I personally care about multiple lifetimes? If from the time I'm alive to the time I die the odds aren't 50/50 for me, then it's not a 50/50, regardless of what you say mathematically. This is a fun one to ponder over different scenarios and really makes you think about real life perspective when relating to mathematics... something mathematicians don't often do =).
Also, I don't really hate casinos and the reason is I remember the old illegal gambling dens that use to exist around town. They were organised mainly along ethnic lines and the police were always raiding them and shutting them down. Today, it is much better as the games are properly supervised in a safe environment. Sure, the odds are with the house in the vast majority of cases and that is the price of entertainment. My local store makes good profits, pays heaps of tax and employs literally thousands of people.
Anyhow, here is the academic paper linked below showing that heads following heads (which is now a restricted sequence) is indeed closer to 40%.
https://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."""
Quote: QFITWhat this paper takes 32 pages to say is that if you use an incorrect method of recording results, you will come to an incorrect conclusion. Measured correctly, the answer is 50% on average.
Yes. Is it me, or was that source paper pretty badly written? Almost like it was written to deliberately bamboozle? It sure got some coverage, which is probably getting monetized somehow.
https://twitter.com/jben0/status/657331384604823553/photo/1
The guys that wrote the paper surely aren't idiots (nor are the people who I've know to make money at roulette). How they do it is in dispute though? To me this topic is interesting. I suspect the same for many who read this site.
Good find at the link you posted that shows the counter argument: https://sinews.siam.org/Details-Page/hot-hands-streaks-and-coin-flips-how-the-new-york-times-got-it-wrong
Quote: OnceDearNot 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.
JUST IMAGINE THE GIGANTIC EDGE THAT YOU WILL HAVE IF YOU BET BLACK AFTER 10 REDS IN A ROW HAVE COME
MY GOD YOUR EDGE MUST BE AROUND 90% WHEN THAT HAPPENS
i can't talk anymore, i've got to go. i'm taking out a 2nd mortgage and then i'm headed out to track red and black on roulette wheels. i won't be back for a few months.
hahahahahahahahahahhahahah
Would you bet that the next coin after any observed head, is more or less (or equally) likely to be a head?
In any coin flip string of 1000 flips (or 100 or 10000 or 100000 etc) circle all that have any of the above patterns? What's the odds that the coin "X" after these specific patterns, will be another head.
For example, H H H "X". Do you think X=Heads = 50%? Why or why not?
Quote: QFITRoulette is beatable. But, it's seriously difficult and has nothing to do with betting patterns. It's also not related to biased wheels. (Problem with biased wheels is you have to analyze the wheel, which will take time, and major casinos regularly rotate the wheels between tables to avoid such an opportunity.) You must look for some unique characteristic of a game to find its weakness. The weakness in Roulette is that it is the only game where you get to place a bet after the game has been put in motion. Think of what that would mean at the track, or so many other games.
You're talking about wheel clocking, I believe. The subject has been talked about here before, so I'm not sure how much you need to beat around the bush aside from specific instructions maybe.
I believe it's possible if you can get your bet in late enough. But from my research on the subject, most newer wheels have changed the design of the ball slots which makes it much more difficult.
Similar to dice control (if possible), there is so much skill and human variables involved that it would be very difficult to mathematically prove an edge.
OTOH, I give no credence whatsoever to dice control.
Quote: bazooookaMaybe others disagree but I wouldn't just call math professors idiots.
The authors are economists, not mathematicians. Economists are famously prone to bad pattern calls.
Quote: QFITI'm talking about Scott's aural techniques, which involve more than clocking as you must also evaluate the consistency of the dealer in both spinning and throwing. Clocking, in and of itself, is useless. And yes, the new pocket designs may have destroyed this opportunity where they exist. (Although there was an opportunity in the UK some years back.)
OTOH, I give no credence whatsoever to dice control.
Right, I've studied the technique but concluded the learning curve is too high, or outright impossible with new style wheels.
I think dice control is largely crap, but maybe has a chance if you can find a casino that is lenient about hitting the back wall.
Even doing it by hand over 100 or 1000 flips and then circling the patterns and tracking will show you there is more tails than heads once you restrict it to picks that have one one or more heads in the first 3 spots of a 4 flip sequence
1. HHHH
2. HHHT
3. HHTH
4. HTHH
5. HHTT
6. HTTH
7. HTHT
8. HTTT
9. THHH
10.THHT
11.THTH
12.THTT
13.TTHH
14.TTHT
Quote: QFIT50%. I'll not bother to explain something that has been known for centuries anymore than I will try to explain why 1+1=2 (in a non-binary system).
.......Quote: bazooookaOnce you restrict the patterns to heads in any of the first 3 spots of a 4 flip sequence the 50/50 will not hold for what follows next. In fact the restricted patterns already have more heads than tails thus we'd expect tails to occur more often, around 60% after a head. I'm sure some excel types can prove this.
Even doing it by hand over 100 or 1000 flips and then circling the patterns and tracking will show you there is more tails than heads once you restrict it to picks that have one one or more heads in the first 3 spots of a 4 flip sequence
1. HHHH...... one heads zero tails......................................... counting the very last toss in the sequence
2. HHHT...... one heads one tails
3. HHTH...... two heads one tails
4. HTHH...... three heads one tails
5. HHTT...... three heads two tails
6. HTTH...... four heads two tails
7. HTHT...... four heads three tails
8. HTTT...... four heads four tails
9. THHH...... five heads four tails
10.THHT...... five heads five tails
11.THTH...... six heads five tails
12.THTT....... six heads six tails
13.TTHH....... seven heads six tails
14.TTHT.............seven heads seven tails...................................................results: 7 heads and 7 tails
say whaaaaat?
quote from Christopher Hitchens:
............."that which can be asserted without evidence can be dismissed without evidence."
.....................................class dismissed
I think the heart of the fallacy is in treating these coin flips as if they're happening in isolation. They're not. You flip 4, then 1st of the second set, it's also 5. And it's 4 for the 2nd flip, overlapping. The truncation is artificial. It is one continuous event, in which all results eventually appear.
You, in fact, have the two arbitrarily excluded, equally probable events in your list of "significant" trends. For example, If #5 occurs, then is immediately followed by #13, the sequence is HHTTTTHH. That sequence includes, in order, #8 HTTT, excluded TTTT because it doesn't fit your theory, and excluded TTTH (ditto).
Each individual occurrence starts its own streak. It is an independent event. And has no relation to or dependence on what happened before. The fact that you can record it means it is in the past, but the next event is in the future. 50/50.
A perfect roulette game is one wherein half the money is bet on Red and half of it is bet on black. The house just uses the loser's money to pay the winner's bets and keeps reaping money because winners are nicked 5.26%.
As far as hurting anyone's head. I suggest that maybe this topic doesn't interest you or others who have said it's idiotic. Anyhow, I think it's been established now that sequences are different than individual outcomes. And a fair bet on any random flip is different than betting on what will happen on a "conditional" flip. The link in the first post explains this better than I ever will. As far as convincing, there is nothing be argued that I know of. Discussion is fun but there seems to be no real disagreement among the posts I've read.
Quote: QFITThe authors are economists, not mathematicians. Economists are famously prone to bad pattern calls.
what you said here is so VERY VERY true. economists love to speculate and intimate that their calls are highly probable when they are not. they use words to make their points, not clear, provable mathematics.
I strongly disagree with your assessment. The article isn’t built on flawed measurement methodology, it exposes an analysis flaw that was overlooked until now. The probable result of a fair coin flip being 50/50 isn’t challenged. In fact, the coin flip example is used because that won’t be challenged.Quote: QFITSorry, but the article is utter nonsense built upon a flawed measurement methodology.
The example illustrates a facet of small sample analysis that appears to have been overlooked: given the occurrence of an event, the probability of a re-occurrence for the next sample is less than the probability of the event occurring in general. This applies to small sample analysis and is in no way useful for predicting when a “hot streak” will start or end.
To summarize what has already been posted in this thread for the coin flip example, 2 of the 16 possible 4-flip sequences yield no result for the given parameters. Of the 14 sequences producing a result, 6 have a 0 probability that a particular head flip will be followed by a head flip, 2 have a 1 probability, 4 have a 1/2 probability, and one has a 2/3 probability. I won’t chew through the math, but the probability that a head flip will be followed by another head flip for a random 4-flip sample is less than 1/2.
The article is about small sample analysis, not prediction. I believe it is only helpful for analyzing event results where the probability for an event can vary and the aim is to uncover that variance. Such as for the sporting events the article cites.
Quote: bazooookaThe only point of this thread was to start a discussion on "hot hand". I don't even play roulette. However I have friends all over the gambling landscape.
As far as hurting anyone's head. I suggest that maybe this topic doesn't interest you or others who have said it's idiotic. Anyhow, I think it's been established now that sequences are different than individual outcomes. And a fair bet on any random flip is different than betting on what will happen on a "conditional" flip. The link in the first post explains this better than I ever will. As far as convincing, there is nothing be argued that I know of. Discussion is fun but there seems to be no real disagreement among the posts I've read.
What???
This topic is idiotic because the content is purely false and your posts are completely ridiculous.
Quote: bazooookaIf we observe a string of 1000 coin flips of a fair coin. And we then pick out any/all sequence of 4 flips that has 1 or more heads in the first 3 spots ex.HHHT or THHT or HHTT or TTHH or HTTH or THHH etc..
Would you bet that the next coin after any observed head, is more or less (or equally) likely to be a head?
In any coin flip string of 1000 flips (or 100 or 10000 or 100000 etc) circle all that have any of the above patterns? What's the odds that the coin "X" after these specific patterns, will be another head.
For example, H H H "X". Do you think X=Heads = 50%? Why or why not?
It was 50% : It is 50% : It will always be 50% right up until the moment it lands and then it will be 0% or 100%
Now stop posting nonsense and links to gibberish. We do understand how someone somewhere has managed to generate ad revenue from one of the sites you've linked to, or maybe has used this sort of publicity to get media to pay for a story. That's the only way that anyone could profit from this complete and utter tosh.
Pack it in please.
I am not one of the "50% religion" crowd. Some people have imbibed too much "i.i.d."Quote: BleedingChipsSlowlyThe example illustrates a facet of small sample analysis that appears to have been overlooked: given the occurrence of an event, the probability of a re-occurrence for the next sample is less than the probability of the event occurring in general. This applies to small sample analysis and is in no way useful for predicting when a “hot streak” will start or end.
In real-life, not some fantasy world, coin-flips are not identical - coins get damaged, they land on their edge, etc...
Basketball players are not machines. If you take a really bad basketball player who freaks out, they can definitely go into a cold streak. For example, I have a hand tremor which gets worse under stress. Question: If you take NBA players in actual games, have they trained themselves enough that their throws can be considered "i.i.d" for practical purposes? Who knows? I'm seen a lot of bad statistics, but haven't looked at the actual analysis in this case. The new software which can track difficulty of throws is an interesting update.
In the gambling world, "hot hand" is possible in non-"i.i.d" conditions; e.g. cards are not perfectly shuffled, cards are collected in a specified order and only shuffled in a limited way, slot machine PRNGs are faulty and produce more/less hot/cold periods than one might expect from an ideal RNG.
I'm going to sound like ZK here, but in real-life some "cheating" exists.
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The area of “streak detection” is one that would be a lot of fun to teach in an introductory probability class.
Consider coin flips in real-life for a “claimed fair coin”.
(1) If you see about 50/50 heads & tails in the first 100 flips, you might lean towards i.i.d.
(2) If you see 100 heads in the 1st 100 flips, a biased coin might be expected….even if the coin was claimed to be fair.
(3) How many heads in a row do you need to shift your beliefs?
As a Bayesian would say, when & how do your change your "a priori" belief that the "coin is fair"?
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People’s natural reactions are not “homo mathematicus”. We use reasoning systems biased in certain ways from evolution.
We are animals designed for the real world, where almost all things are not "i.i.d".
We are designed to identify things which are out of the ordinary and will hurt us (e.g. cheating).
So most people have very little natural sense of true “i.i.d.” (e.g. people think a little bit anti-clumped is random). People who learn a little math become "50% religionists" in an attempt to overcompensate for natural reactions.
Finite sequences are more common than infinite sequences for everyday experience. In blackjack or video poker, it takes a huge number of hours to reach N0 or any version of “long-term”. Beginners with a little math knowledge are less familiar with short-term fluctuations (whether dealing with RoR “Risk of Ruin” or Kelly optimal betting).
*** The recent papers are dealing with finite sequence effects. ***
A cool exercise is looking at the expected loss a BJ player with x% edge will see before becoming profitable. Or the flip side, the expected win a regular BJ player will see before heading into long-term loss.
Another cool area is zero-crossings. The behavior of coin flips (or Bernoulli trials) is mostly to stay in long hot streaks and long cold streaks. The expected absolute distance from 50/50 increases over time, although the percentage difference from 0.5 is expected to tend towards zero (one might call this a real-life infinitesimal - the time spent at exactly 0.5 also decreases towards infinity).
Heck, I'm not even a math guy but I can (and you should) see why when there is H,H,H,X - X is not a 50/50 proposition for H or T when we limit our pick sequences with a "condition" to "how often a head follows a head" "after any of the preceding 3 tosses that at least 1 head" in its discrete 4 toss sequence.
***Read Bleeding's post on page 4 or the original article until that makes sense to you.
I must say that "knowing people who've done quite well" at roulette is amazing; I'm very impressed. But I do wonder just how one might have determined that these other people have done so well. And given what a small world that'd be, whether it might include anyone we've come to know here.Would any of them happen to have previously done well at divining the patterns of the machines on the casino gaming floors, before taking up roulette?
Quote: BleedingChipsSlowlyThe article isn’t built on flawed measurement methodology.....
To summarize what has already been posted in this thread for the coin flip example, 2 of the 16 possible 4-flip sequences yield no result for the given parameters. Of the 14 sequences producing a result, 6 have a 0 probability that a particular head flip will be followed by a head flip, 2 have a 1 probability, 4 have a 1/2 probability, and one has a 2/3 probability. I won’t chew through the math, but the probability that a head flip will be followed by another head flip for a random 4-flip sample is less than 1/2.
I'm not exactly sure what the point of departure is but I see a lot of talk about flipping coins. As I've been saying for 20 years now, assuming a fair coin, the probability of every flip is 50/50 and it doesn't matter what the pattern of heads and tails was preceding it. This also goes for dice, roulette, keno, and all games of independent trails.
As it relates to sports, that is getting more subjective. I tend to think that streaks and momentum are overrated by squares handicapping sports. For example, if I heard a tout saying that the Patriots have a lot of momentum because they have beaten the spread the last several games, I would be inclined to bet whatever the other team was in the next game, if forced, because I know a lot of squares will put too much stock in the momentum theory, creating value the opposite way.
It really bothers me when announcers say things like "Adam Vinatieri has missed his last three field goal attempts from the 50-yard-line." Who cares about sample size of 3? I'd rather hear his percentage the last three seasons or so.
I think most of us here are talking past each other. I still contend that there are event sequences where heads will be less than 50%.
Since there are "16" 4 flip sequence and only "14" has at least 1 head in the first 3 spots thus 2 sequences will be excluded. Of the 14 left, 6 have a zero probability that a head will be followed by a head and the others have different degrees of probability that "head will follow a head".
It all adds up to around 40% as shown in charts linked below. Flip sequences have different odds than individual flip outcomes.
Agreed there is no gambling edge to be found here..
https://sinews.siam.org/Portals/SiNews2/EasyDNNNews/thumbs/383/117HotHandsTable.JPG
https://pixel.nymag.com/imgs/daily/science/2016/08/12/12-coin-flip-probabilities.nocrop.w710.h2147483647.gif
Point is that sports (and I look at sports only through the lens of sports betting) doesn’t relate well to gambling games where the instruments (e.g. cards, wheels) don’t have emotions; and an accomplished AP isn’t supposed to have emotions.
Quote: bazooookaBut in all "16" 4 flip sequence only "14" has at least 1 head in the first 3 spots thus 2 sequences will be excluded.
Nope. The study is fatally flawed as it uses a technique of recording that is silly.
I'm a sports bettor too and I'll stick to that. I'm not sure how people (and I know some) make serious money at roulette.
That being said I do believe in "hot hand" for basketball. And this study linked in the original \post does have implications for online betting that does allow one to bet on whether a player will make/miss his next shot.
Nevertheless, the approx. 60% occurrence of tails following heads under "restricted scenarios" and sampling as shown in links a few posts above, do seem sound. If not; what do those graphs show?
Quote: bazooookaIf not; what do those graphs show?
"There are three kinds of lies: lies, damned lies, and statistics." --Benjamin Disraeli
One must be extremely careful when presenting stats as misapplication of statistical theory can result in absurdity.
