The question isn't whether or not information can start in a vacuum, the question is whether or not knowing how the information was generated necessarily affects the probabilities.

I wouldn't say necessarily affects- you can certainly generate some information in many different ways that all produce the same probabilities- but it certainly CAN affect the probabilites (think of ways to generate a number 1-6 with a die and various algorithms for re-rolling- some matter and some don't).

cute trick, but there's not an equal number of sets starting with H and T. there's 8 H and 6 T. you can't arbitrarily exclude TTTT and TTTH. that's where the missing parts are.

reminds me of counting eleven fingers. 10-9-8-7-6 on one hand and 5 fingers on the other hand makes ELEVEN!

That's not where the bias comes in. If you looked at all Heads flipped in the first 3 flips (which would also exclude TTTT and TTTH) and tallied the percent of Next Flips that were Heads you would get 50%.

What their method does is to just take 1 random "First-Three" Head from each sequence and tally the Next-Flip result for those Heads. The effect of this method is that HHHH counts no more in the calculation of their probability than the sequence TTHH. In other words, in their method 3 successful Hot-Hand flips in HHHH count no more than the 1 successful Hot-Hand flip in TTHH.

PairTheBoard

Maybe the coin isn't perfectly balanced and the etchings on the tail side make it just a little more heavy than the heads side?

I guess something like that was the idea. I could not duplicate the result shown on TV though.


PairTheBoard

Have you tried something like this?

http://mathtourist.blogspot.com/2011/02/penny-bias.html

Edit: Oh wait, there might be a bevel to the coin to produce the effect. But here's a science-news article:

http://web.archive.org/web/200803140...40228/fob2.asp

Yea, this is the claim I tested. I tested both the old wheat tail and memorial tail types. The test was by spinning 100 pennies. I think I repeated the test twice for each penny type. All the results were in the normal range for 50%. Practically nil probability that the 80% could be right under the conditions of my test. I spun them vigorously on a smooth kitchen table.


PairTheBoard

That opening to the statement was just prefacing.

And it's where, not how.

Yes, its true that you do play differently when you have the "attitude of a winner." Sometimes, you play confidently, which leads to more success. And sometimes you play overconfidently, which leads to less success. And there is no reliable way to predict which of these you (or the Knicks point guard) will do from situation to situation.

So, sure, it "exists" if thats what you want to believe, it just doesnt matter.

Right, we "know" this. Of course, we "know" lots of things that are false, but its ok, lets just say its true. We also "know" that chuckers like Kobe will take terrible shots when they feel they are "in the zone" and we know that overconfidence can lead to failure in all sorts of scenarios. And we ALSO know that for some reason, hot streaks arent just runaway positive feedback cycles.

So.....where does that leave us? Oh thats right, the "hot hand" is a lot like God. It exists as long as we dont try to actually find it, or use it to make predictions about the real world, or in any way expect any useful thing from it ever.

Its possible that some people believe "the zone" doesnt exist. But its a silly belief to have. I mean, it might be true, it probably is, but there is no reason to have to defend such a strong version of the idea.

Just say that it has no predictive value, and "the anti-zone" is just as prevalent, and players switch in and out of each at will. Much easier to defend.

That is unless you two are fabulously wealthy sports bettors or something, since this "zone" thing seems like an amazingly profitable concept.

Do you think a basketball player who hasn't taken a shot in a week has as good a chance of making a free throw as the same player who has just taken twenty practice shots? If not, why?

You sound like someone who has no idea what he's talking about. Why don't you take a moment to describe what you think "being in the zone" is and how one could profit off of it.

Sorry if this point has been made in this thread already, but this is just basic statistical regression....
Take a simplified version of the original problem...a fair coin is flipped 100 times: Our prior expectation is that we would end with 50 heads 50 tails...
But if after the first 50 flips we have recieved 40 heads and 10 tails... Our 50 in 100 model is looking unlikely, but was still accurate at point on inception. The expectation then would be that in the next 50 spins, we should regress back to the normal and find we get 40 tails and 10 heads....
This analogy using 100 spins is obviously quite skewed: but when you look at spinning the coin 4 times, you expect 2 heads 2 tails.
After the first spin comes heads, our expectation is to now spin two tails and one head....
If we continue this process a million times our overall stats will regress to a perfect 50-50 distribution...by then only selecting the spins following a head we will see a regression back towards tails. Even though the probability of each individual spin is still 50-50....

To extrapolate to a wider example, take a 37 wheel roulette wheel, if we look at 37million spins which we would expect an even distribution over, you can expect a pretty close to perfect Million of each number, if 'zero' came up 750,000 times in the first 5 million spins, statisticaly you would expect the number distribution to regress back to 1million of each so statisticaly zero is less likely to spin, even though on each individual spindle chance is still 1:37.....
I hope this makes sense

Well, I think it was common knowledge that zombies didn't have functional brains, but now we have evidence that necromancers don't either. Neat.

I believe this is the easiest way to see what's happening when the OP method is used for repeated trials of 4 coin flips.

The "regression to the mean" idea is about as wrong as you can get. Or ... what Tom said.


PairTheBoard

When I was a kid I used to think that if you spun a coin with heads facing you, it was more likely to come up heads, and vice versa. You can certainly cheat by leaning the coin away from you slightly and give it a weak spin. Good for betting games against other 10 year olds. I never tried this, but if you really want to cheat, machine a barely noticeable bezel onto the edge. Maybe someone with access to very sensitive equipment can place normal coins as vertically as possible and see if there is a force in a preferred direction.

Stand a bunch of pennies (20-30 should be enough) on their edge on a table facing a bunch of different directions. Slam your fist on the table to cause the coins to fall. Report your results.

I see the WSJ is trolling math clubs around the country.

Bumped from the future.


PairTheBoard

I have a coin flipping question:

As you flip more coins, the average ratio of heads vs tails approaches 50%, correct?

But otoh, as you flip more coins, the probability of flipping exactly 50% heads and 50% tails decreases (I believe it halves every two flipped coins, starting with thre highest with 2-coins at 50% likely, then 4-coins at 25% likely, 6-coins 12.5% likely etc).

In which case, isn't it correct to say that as n approaches infinity, the probability of flipping exactly 50% heads and 50% tails approaches zero?

And yet, we say the % heads vs. tails approaches 50%. I can only think it's because all the non-50% combinations of heads vs tails balance each other out?

Bonus question:

How would you try to convince someone that thinks "if I flip 99 heads in a row, the next flip is more likely to be tails, because the Law of Large Numbers means coin flips converge on 50%".

I was surprised to hear it from this person, but at the same time, I was more surprised at how difficult it was to find a persuasive explanation, instead of just declaring "every expert resource calls this the Gambler's Fallacy", which is not particularly persuasive.

Telling them that each coin flip is independent hasn't changed their mind, I think it's because they are imagining that the "convergence to 50%" is changing the probabilities over time, I think they are viewing the Law of Large Numbers as not simply descriptive but prescriptive somehow.

It might just be an intuition they have that I can't see how to change, but it's been interesting to talk about. They are otherwise very smart and rational, but not familiar with gambling.

Yes to first question.

The "non-50% combinations of heads vs tails balance each other out" observation is true for any n and why the EV for any n is n/2. The reason the % heads vs tails "approaches" 50% is because for n large the number of ways the flips can turn out to get k heads (or k tails) becomes far more numerous for k close to n/2 than for k further away.

As far as seeing 100 straight heads flipped. If I saw 100 straight heads flipped I would conclude a rigged coin is being used and barring a switch out I'd bet on another heads. However, assuming a fair coin for purpose of theoretical argument, I'd point out that the coin has no memory and after a trillion more flips the first 100 will have close to zero effect on the average. After another trillion flips the average will very close to n/2 with very high probability regardless of what happened in the first 100 or 101 flips. That's what the law of large numbers says.


PairTheBoard

My understanding of conditional probability is limited to the formula P(A given B) = P(A and B)/P(B).

Here, B seems clear enough: at least one of the first three flips is a head, so P(B) =7/8. But then A seems to be in a different event space, and a more complicated formula than the above is being applied. Is that correct?