I'm sure they're out there, and I'm sure they're wrong.

There's a reasonably good analogy to winrate. Your winrate is not some fixed value. It changes with the opponents, your mental state, and your opponents' mental states. There are some moments when your brain is firing on all cylinders and you're reasoning very clearly, and you make good decisions and have a higher EV. Other times, you're frustrated or tired and make bad decisions and have a lower EV.

Why is so hard to believe that a physical activity like basketball isn't subject to the same mental variations?

Won't bother to read this thread unless someone is saying that a flip is more likely to be heads if the previous one is.

It is true as far as free throws is concerned though. Obviously.

The WSJ article is about a published paper which tries to give the impression that a coin flip is LESS likely to be heads if the previous one was.


PairTheBoard

Little fun problem;

Test hot hand hypothesis or rather a connection of outcome to prior ones by studying the standard deviation and expectation of the sequence instead (and comparing with the true binomial 50% case).

For example lets say i ask what is the sd of a truly random 50-50 each run independent from the ones before (say +1 and -1 50% each). Thats an easy result.

Now imagine you have a system that after a +1 has probability p to be +1 that is p>1/2. Lets also say it has a probability q>1/2 to lead to -1 after a -1.

What is the expectation of that sequence of say N trials if it starts at 50% +1 or -1 on the first trial as function of N,p,q. What is the sd of N trials?

Basically compare the observational details of Markov chain with the true Binomial without looking for conditional probabilities. It might save time for example or something or prove easier to test.

Coin "in the zone"


I'm waiting for a serious investigation whether there is even a slight difference between the frequencies of heads and tails for specific coins. Maybe the heads side is a bit heavier, increasing the frequency of tails?

Let's design a machine. 10,000 plastic boxes with the same coin type in each, arranged in 100x100 on a platform which is shaken by a motor. Optical reading of the outcomes of each coin. Shake it 100,000 times, can be done in a week or two. You are getting one billion outcomes totally. Will there be a statistically significant difference? I guess there could be something of 50.00001% for either side?

I always thought with respect to this puzzle that the way you found out that at least one child was a boy is that you were told, when you were told the puzzle, that "at least one child is a boy." Does that not avoid the potential ambiguity?

I think it depends on how the information you receive was generated. You can read the Wiki link in the section "Analysis of the Ambiguity". If the information you receive was generated by someone looking at both children and seeing that at least one is a boy then the answer is 1/3.

However, according to Wiki, that information could be a "random" true statement in which case the answer is 1/2. This is the first I've heard of the concept of a "random" true statement. You can see what Wiki says about it. I'm going to have to think about it.

It's at least easy to see that the information could have been generated by someone who just looked at the older child and happened to see he was a boy. He then gave you the watered down information that at least one child is a boy. If that's how the information you received was generated and you guess "the other child is a boy" you will be right half the time.

I think it's certainly the most natural assumption that the information was generated by someone looking at both children.


PairTheBoard

Looking at the Wiki "Analysis of the Ambiguity" I think this is the alternative scenario by which the information "at least one child is a boy, (ALOB)" was generated. The information "at least one child is a girl" is denoted ALOG.

First a random family is generated. If it's GG the statement ALOG is generated. If it's BB the statement ALOB is generated. If it's BG or GB then in either case a coin is flipped. If Heads, ALOB is generated. If Tails, ALOG is generated.

If the information "at least one child is a boy" that you receive was generated this way then the answer is 1/2.


PairTheBoard

I think this is pretty good confirmation that the most natural understanding of the question gives the answer 1/3.

In 1991, Marilyn vos Savant gave a version of this in her popular column.
From Wiki -
--------------
In response to reader response that questioned her analysis vos Savant conducted a survey of readers with exactly two children, at least one of which is a boy. Of 17,946 responses, 35.9% reported two boys.[10]
=========

Slightly more than 1/3 because it's not coin flips.


PairTheBoard

Oy. That's a terrible way to justify the answer.

Why?


PairTheBoard

Struggling to see how this is relevant. If you don't know that the information was watered down, it seems simply incorrect to say that P(other child is a boy) is anything other than 1/3.

Bar-Hillel, Falk and Nickerson weren't frequentists, were they?

It seems to me the same objection could be made to what's evidently the standard alternative scenario described in Wiki where the "at least one child is a boy (ALOB)" is considered a "random true statement". That means a random family is checked and if it's BG or GB they tell you "at least one boy" half the time and "at least one girl" half the time. This is just one of those times they tell you "at least one boy". It's 1/2 in that scenario.

We'll see why Aaron W. objects to the statistic. But it seems to me the natural way of understanding the question is to look at this as if it were a random 2 child family drawn from those with at least one boy (under simplifying assumptions about 50-50 and independence).


PairTheBoard

It sounds like the 1991 version of an internet survey.

Certainly not the best statistical method but I think it's the idea that matters. And evidently the method didn't bias the result too much because it's just about what you'd expect considering boys are more common and there's a little correlation with sibling gender.

PairTheBoard

You appear to have more information here than being told "at least one boy" without being told how that was ascertained, so why wouldn't the probability be different?

It seems to me that if you are only told "at least one boy", and nothing else, how "at least one boy" has been ascertained (by the person telling you) is relevant to the frequentist but not the Bayesian.

Yea, that sounds exactly right and strikes me as rather insightful! If whatever experiment generating the information is repeated numerous times the average result will depend on the generating method. But as far as updating the available information in this one instance the Bayesian updates it according to the standard conditional probability, i.e. P(both boys | at least one boy) = 1/3.


PairTheBoard

Right. But let's say that the number came in at around 50%. Would we question the analysis or the data? So it seems like we're only looking for something that can confirm what we already believe.

(I'm also reminded of analyses of Mendel's work that suggested that he faked the results because they were too good. Since this one is far enough off from the theoretical value, it's plausible that it was a good sample. But it still has that feeling of being a very biased approach to the data.)

What's 40%?

It was a biased approach to the data. Census data would have been better but probably not as much fun for Marilyn's readers.


PairTheBoard

I must be thinking of this naively, but I'm wondering why any assumptions at all should be made, or need to be made, about how the information was generated. Why can't the person presenting the puzzle simply stipulate that "at least one is a boy," and by that stipulation imply that no assumptions should be made about how the information was generated? Or does that stipulation then in fact implicitly bring the ambiguity into play?

To put the point another way, I want to ask why there would be any reason to interpret "at least one is a boy" as having been generated by first selecting a family, and then making a random, true statement about the gender of one child in that family (which then changes the sample space and so the probability that both are boys) if we are simply stipulating that "at least one is a boy"?

Right, but why would there be any reason to make the assumption that that's the way the information was generated in the first place?

Is the problem somehow ill-defined unless we specify a way in which the information was generated?

I guess that's the idea. Although it seems somewhat telling that the examples they give produce the answer of 1/2 which so many people think it should be for the wrong reasons, i.e. nothing to do with the idea that it matters how the information was generated. With such a biased method of generating the information you can produce any probability from 1/3 to 100% if the method is known, or if the experiment is repeated numerous times to reveal the bias. I suppose it's accurate to say the question is not perfectly specified but it seems rather pointlessly nitty to me. I suspect you could nit up most probability word problems in similar ways. (e.g. How do they know the coin is fair? or How do you know they don't only tell you the first coin came up heads when they know the second coin came up heads as well? How did they get this family to begin with?)

I suppose this question of how the information is generated has some motivation in that there are other versions of this puzzle where there is additional information in the puzzle about how you conclude "at least one child is a boy" and the way the puzzle reveals that information is relevant to the calculated probability. But in those cases the additional information is available to you.

After all, probability models are all about handling incomplete information. The probabilities they generate will generally not be proved by frequencies for repetitions of an experiment about which the model has incomplete information. If the model had a perfect specification for the experiment the model would produce exactly one outcome with probability 100% (barring quantum effects).


If the only available information is "at least one boy" then it seems to me the only reasonable way to update the probability model based on the available information is to rule out GG leaving you with the equally likely outcomes of BB, BG, GB. That's certainly what a Bayesian would do looking at probability as an information theory and probability models as encoding the available information.


PairTheBoard

Information doesn't start in a vacuum and isn't linear. It's an oscillatory process.

Take a sapient civilization looking for a deity.

a priori nihilio.