A Rejection of Sklansky

[PairTheBoard]

David made the following statement on his "What Question?" Thread. That thread is about the meaning of his Probability Statements. If you don't know what he means by Probability you should read that thread. His is not a Frequentist Definition. This Thread is about this particular statement of his. I believe it needs to be Greatly Discounted.

Link to "So What's This Problem I Won't Answer?"
[ QUOTE ]

Sklansky -
If the only information you have is the number of choices, your personal probability should be equally divided among them. Perhaps because you have data that shows that when the only information is the number of choices the choices have come up equally. But in this particular case even if you don't have the data it is silly to say that you are falling back on subjective judgement. Rather you are falling back on not being a moron.


[/ QUOTE ]

I'm not Rejecting his statement. Rather, I am Rejecting his contention that people are not aware of this principle and he must preach it to them. That is exactly NOT the problem. Not only are people well aware of this principle. They are Too aware of it. In fact they are so aware of it that they apply it habitually without even thinking. The way they apply it in practice amounts to the very illogical principle,

"Whenever there are two Choices they must be Equally likely."

This causes people much much more problems in their thinking than not being aware of Sklansky's principle. And after all, Sklansky's principle in its pure form is totally Theoretical. You never have two Choices where the only thing you know about them is that there are two of them. If that's all you knew, you wouldn't be able to distingish which was which. That never actually happens. Only in a theoretical state of imagination can we imagine two choices without being told what they are. Once we are told what they are we know something More than that they are just two Choices.

The problem is that people in general, after being told what the two choices are, have a false intuition that the two Choices are equally likely. If they have any difficulties seeing any implications for their relative liklihood from the description of the choices they automatically assume they are equally likely. They need to break that habit. They hardly need Sklansky telling them to apply it more freely.

Some examples from my exerience on these Forums and with students I've taught.

1. The Monty Hall Problem. There is a prize behind one of three doors A,B,C. After you pick a door, Monty Always opens one of the remaining doors and shows that it is empty. You know this. He then gives you the option of switching to the other unopened door. What is the probability the other unopened door contains the prize?

Many Many Many people automatically say 50%. Why? Because they are Not aware of Sklansky's Principle? No. They are Too aware of his principle. They are so aware of it they assume it applies here because they can't see any reason why it shouldn't apply. They are so convinced of this, that even after they are shown the logic of Conditional Probability (Bayes) as it applies to this problem, they remain convinced Sklansky's principle applies and they stick to their 50% assertion. Their conviction about using Sklansky's principle is so strong it Prevents us from teaching them to use Bayes' Theorem. They hardly need to be preached to by Sklansky that they are not using his principle enough.

2. The Two Envelope Problem. There are Two Envelopes. Dollar Amounts have been chosen in some way we know nothing about whereby one Envelope contains twice as much money as the other. You pick one at random, open it and see $100. You can either keep the $100 or switch to the other envelope. What is the probability the other envelope contains $200 and what is your EV for switching?

A Huge number of people will without blinking apply Sklanky's Principle immediately and say the probability is 50%. Even after consderabale thought they will say 50%. Even after they have been shown the logical contradictions from the resulting EV calculations they will still say 50%. Even after extensive Baysian analysis is shown to them they still think it should be 50%. Why? Because they are not aware of Sklansky's Principle? Sklansky's Principle is so engrained in their psyches they can't prevent themselves from applying it despite the abundance of evidence showing them they shouldn't.

3. Almost any situation where a Baysian analysis would be beneficial. Here Sklansky is trying to get people to take his Baysian approach to probability while at the same time preaching a Principle which only holds for sure in an imaginary Theoretical Setting and which is exactly the thing that Prevents people from taking advantage of his Baysian approach to Probability.

Case in point:

Disease testing. A TB test has 90% accuracy both for Positive and Negative results. Those who have TB test positive 90% of the time. Those who don't have it test Negative 90% of the time.

A teaching applicant must be tested. So he had the test done and it is Positive. He concludes he almost certainly has TB. Why? Because intuitively he is applying Sklansky's Principle. He figures he has two choices. Either he has TB or he doen't. So he applies Sklansky when looking at the TB results. He might just say, he has half a 10% chance of having gotten a False Positive so 5% chance he doesn't have TB. Or he could apply Bayes' Theorem,

(.1)(.5)/[(.1)(.5)+(.9)(.5)] = 5%

to see a 5% chance he doesn't have TB.

Sklanky's Principle is so engrained in people's intuition that it affects their judgement on many many things.

The Extension of Sklansky's Principle to multiple disjoint events where nothing is known except the number of the events.
==================

4. Elementary Probability Lessons
When teaching classes in Elementary Probability it is very easy for students to grasp models where there are atomic outcomes, all of which are equally likely. But when you pass to situations where the atomic outcomes are in the background and it's not clear where the atomic outcomes are in the Events being desribed by the model, the students begin to have difficulties. The reason is that they want to continue applying Sklansky's Principle to the Model of Events which contain hidden atomic outcomes. If they see there are N disjoint Events of this type which they don't have a feel for, they want to think of them as equally likely.

Case in point.
Assume the chance a random child in random family is a boy is 50% for our population. Now,

A family has two children. There are 3 Events. E1=both boys, E2=both girls, E3=one of each. We are told that one of the children in this family is a girl. What is the probablity the other one is also a girl?

Students think it should obviously be 50%. But then they figure they should look at the three Events that were given. Maybe the three events apply and they should use the conditional probability they've been learning. So they think Sklansky to themselves. They see 3 equally likely events. One of them has no girls in it so they can focus on the remaining two that do. They see that E2 and E3 are therefore Equally likely. One is the two girl event and one is the boy-girl event. So even applying conditional probability on the three events they conclude the probability it is a two girl family given that one child in the family is a girl must be 50%. This agrees with their previous intuition and they are done thinking. Sklansky has led them astray again.



For anything remotely tricky like this where it's not easy for people to see why events wouldn't be equally likely they will quite stubornly hold on to the Sklanksy Principle telling them they are equally likely. Even after being shown why the Events are not equally likely. Sklansky's Principle is hardly something people need to be made aware of. What they need to be aware of is the fact that in Reality it is skewing their judgement on many many things.

PairTheBoard

[TomCowley]

Saying that the situation is "only theoretical" is your standard nihilist irrelevant nonsense. "An event will either happen or it won't. What is the probability that it will happen?" You can draw absolutely no bias from the description, any more than you could if you were told there were two outcomes, frumplesnort and dingleberry, described in a language incompehensible to you, and you were asked for a probability. ANSWER the question as it DOES exist, don't say it can't exist.

Monty Hall is a MISapplication of sklansky's principle. If you couldn't assumethat all 3 doors were equally likely to begin with (skalnsky's principle), then you couldn't solve the problem at all.

Two Envelopes is not a mathematically valid problem. When converted into a mathematically valid problem using limits, guess what the odds asymptotically approach? That's right, 50%.

The third and fourth cases are simple math ignorance about how to calculate conditional probability (if shown this problem from a frequency standpoint, even relative idiots can understand the process, although they'll be shaking their head at the answer). It has nothing to do with Sklansky's principle, and is certainly no sort of counterexample to it. If you list the FOUR indexed outcomes, boy-boy, boy-girl, girl-boy, girl-girl, even idiots should get this right.

Some people are godawful at math. DS has nothing to do with this. Please tell me where you teach, so if I ever have kids, I can make sure they don't end up in one of your classes. The last thing I'd want is somebody who can't analyze trivial situations and instead has to resort to "we just have to know something about the outcomes by what they are" gobbledooygook gibberish.

[PLOlover]

[ QUOTE ]
The problem is that people in general, after being told what the two choices are, have a false intuition that the two Choices are equally likely. If they have any difficulties seeing any implications for their relative liklihood from the description of the choices they automatically assume they are equally likely. They need to break that habit. They hardly need Sklansky telling them to apply it more freely.

[/ QUOTE ]

so you're saying the bent coin problem is a specific case of an erroneous general principle, but that in this particular case it happens to (accidentally) be correct?

Or are you saying (more correctly in my opinion), that the bent coin problem appears on the surface to be an application of an erroneous general principle, but upon further analysis has actually nothing to do with the e.g.p., but still you want to warn people about the dangers of the e.g.p.

[PLOlover]

just to clarify, the sklansky bent coin example specifically states that the two choices (heads, tails) are not equally likely. I mean, it's a bent coin. The bent coin example works no matter what the degree of bentness is, or no matter what the ratio of heads/tails is when you flip bent coin.

Is that in dispute?

[NotReady]

[ QUOTE ]

If you list the FOUR indexed outcomes, boy-boy, boy-girl, girl-boy, girl-girl, even idiots should get this right.


[/ QUOTE ]

OK, I'm an idiot, but this is driving me nuts if the chance isn't 50%. The set-up stated 50% probability of a boy, which means 50% probability of a girl. Unless the fact the first child was girl indicates a bias towards girls, I don't see how it can't be 50%. But then, I missed the Monty Hall Problem the first time, though I do get the explanation.

If it isn't 50% how is that different from say a poker situation. I miss 10 flush draws in a row. Isn't my chance of making the 11th flush try the same as the first?

If someone answers this and I'm wrong, please explain the difference in the two situations.

[David Sklansky]

Surely you realize that I'm not even reading the posts on this subject. So I would appreciate it if you don't label them in such a way that I think it is a post about Brandi which I AM reading.

[PLOlover]

[ QUOTE ]
Surely you realize that I'm not even reading the posts on this subject. So I would appreciate it if you don't label them in such a way that I think it is a post about Brandi which I AM reading.

[/ QUOTE ]

now thats funny

[yukoncpa]

Hi PTB,

It seems like there are now three different threads regarding your discussion, and I didn’t follow along on the first two and they got huge, so pardon me if this has been answered.

I can figure out the Monty Hall problem, but I can’t figure out the envelope problem. Before I look in the envelope, the chance of picking the one of two envelopes that has the most money, must be .5. Now it looks like, once I peak into the envelope, my peeking somehow changes the probability of the distribution of money in envelope number two into something other than 50% $50 and 50% $200, because, I don’t accept that I’m going to make money by doing a switch. If that were the case, I could make money from nothing. In other words, suppose you and I both chose an envelope. Now we both come to the conclusion that switching is correct, so we switch with each other, and we both make money. This would be nice, I’d be happy to switch envelopes with you all day long. But I don’t see this as working any more than I see cockamamie crap systems as working.

Can you explain how before the fact, there is a 50% chance of choosing the envelope with the most money. But after peeking, evidently there can not be a 50% chance that the remaining envelope either contains half of your amount or twice your amount, else you would create black magic. What’s the answer? If it’s already been answered on one of the two other threads, just tell me so and I’ll search some more. Somehow I never found the answer and decided a lengthy letter would get me a faster response.

Thanks (also, I’m being a bit humorous, I’m sure, peeking has nothing to do with it, but rather some sort of mathematical trick that I’m unaware of )

P.S. I see TomCowley has said that the envelope problem is not mathematically valid. I accept this using common sense, but where is the invalidity?

[TomCowley]

The invalidity is that there is no such thing as a uniform distribution over an infinite number of items. If you structure the problem in a way that probability theory can handle, taking limits as the range goes to infinity (as I did for a variant in the Probability forum- check it out), there's always a chance you're opening the "biggest" envelope, and switching it gets very very bad (enough to make switching 0 EV overall, despite the gains from every other switch). The odds of not gaining by switching approach 0, the odds of the other envelope being double approach 50%, but the EV stays 0. I confused myself for awhile working through it until I realized what I'd done wrong.

[Siegmund]

I agree with PTB's contention that this "no information, assume equally likely" principle is overused, not underused. Come to think of it, not sure I've ever met anyone who DIDNT subscribe to that notion in some form.

Not sure how that is a rejection of Sklansky's point - just an observation that "no information" is a situation that's a lot less common than a lot of people believe.

Bit disturbing that he doesn't read the threads he starts, though.