Bear’s Bastards


No man will be found in whose mind airy notions do not sometimes tyrannize, and force him to hope or fear beyond the limits of sober probability.

~Samuel Johnson


"Illegitimis non carborundum."
Lat., "Don't let the bastards grind you down."
Gen. Joseph Stilwell






What up all you Peaches and Pineapples out there. What up all you bear cubs. For reasons I am not able to fully explain it is International Take Your Sex Slave to Work Day. (Schrodinger, like there'd be any doubt.)

Over on Monkey Barn we found a hilarious list of 100 Reasons Why You're Still Single. And I have my third set of gift ideas for you to consider getting Hyperion for Hyperion Day, or just anyone you care about. (But seriously: me.)


$$$$$$$$$$$

[Yesterday's column generated a fair amount of Comments and Emails, causing me to push today's plan until tomorrow to write another probability column. To those of you who could not follow yesterday, I promise today will be easier. Illegitimate children will run free! What more could you ask for? However, it might help to read yesterday's column before tackling today's. Thanks.]



#463 Bear's Bastards


At the risk of scaring all of you under the covers, we are going to delve into probability once more.

Before we get there, though, I want to explain the writer's dilemma. Often a subject will come up that calls for a technical discussion. I assume at least 68.4% of you are fairly bright, or else why would you show up here day after day? I do not want to talk over anyone or down to anyone. On the other hand, being bright and reasonably educated does not mean training in all different fields. If I went to a website on knitting and found purl jargon, I would have no clue.

What I am trying to say is that I wage a constant battle between the way I might talk to a friend and keeping my columns readable to a wide audience. Sometimes that means condensing the logic, math or history of a subject, to keep people interested. Moreover, there are times when my storytelling style is inexact and causes the more technically minded out there confusion.

In yesterday's column, I talked about three packages of dried fruit, and used an analogy of the Monty Hall question to try to illustrate the probability of a certain set of circumstances. Rather than illuminate I ended up confusing some of you who felt the two situations were not similar enough to compare. They were, but my lack of specificity might have made that difficult to see, and if so I apologize.

(For those who do care about the technical details, I did actually knowingly take one peach, and I actually offered the choice of "switching" to past-Hyperion in the closed loop. Do you see why I left that information out? I was trying to keep the more regular folk from bleeding out of their eyes.)

The problems of understanding probability come down to how counter-intuitive it often seems, how the answer just does not "feel" right. Secondly, it is difficult for us to understand a closed loop and what information falls into that and what does not. I have a different example today; one I am hoping will make things more clear. Then again, I spent almost three hours arguing with friends about this on IM last night, so who knows. I call this problem Bear's Bastards.

For the following Situations, we are going to stipulate that the probability of any given birth is 50/50 BOY/GIRL, regardless of how many BOYs or GIRLs are already in the family. (I know there is data to suggest otherwise, but just ignore all of that, as it does not pertain to the actual question.)


Bear's Bastards - Situation A

I find out that Bear has two kids I never knew about. (I guess all those times he was "hibernating" were lies.) I have no knowledge about these kids whatsoever. I find out from Koz that one of the kids is named "Hyperion." (We are going to stipulate that means it is a BOY, otherwise I might cry all night and never get this written.)

Anyway, here is the question: what is the probability that the other child is a GIRL?

Answer - 2/3

I know, it is the dreaded 2/3 again, but hopefully this time we can make it work for you.


As you know, there are four possible ways Bear's kids came into the world. I have added names to help keep track of all the bastards.

Outcome #1 – Older Child BOY (Hyperion); Younger Child BOY (Gareth)

Outcome #2 – Older Child BOY (Hyperion); Younger Child GIRL (Swan)

Outcome #3 – Older Child GIRL (Swan); Younger Child BOY (Hyperion)

Outcome #4 – Older Child GIRL (Swan); Younger Child GIRL (Moiraine)


Every single person reading this, no matter how much probability hurts their head, understands the four ways two kids could have come into the world. Joe produced either BOY-BOY, BOY-GIRL, GIRL-BOY or GIRL-GIRL. We cool so far?


I have been told one of the children is Hyperion, meaning a BOY. This means that one of our outcomes is no longer possible, giving us just three remaining:

Outcome #1 – Older Child BOY (Hyperion); Younger Child BOY (Gareth)

Outcome #2 – Older Child BOY (Hyperion); Younger Child GIRL (Swan)

Outcome #3 – Older Child GIRL (Swan); Younger Child BOY (Hyperion)

Outcome #4 – Older Child GIRL (Swan); Younger Child GIRL (Moiraine)
NO LONGER POSSIBLE


Of the three possible outcomes, in two of them the "other" child is a GIRL. Outcome #2 gives us Swan, as does Outcome #3. Only Outcome #1 gives us baby Gareth. Therefore, the probability of Bear's other kid being a GIRL is 2/3.


Got that? The reason the probability is 2/3 and not ½ is because we are not determining the probability of each individual birth, but of the possible outcomes that could have taken place. Once we added the partial information, it is straightforward to figure out.

Now, are you ready to make it more complicated?


Bear's Bastards - Situation B

Let us assume that I found out the OLDER child is Hyperion, a BOY. What is the probability that the younger child is a GIRL?

Answer - ½.


How can this be? Again, all we have to do is look at our four possible Outcomes:


Outcome #1 – Older Child BOY (Hyperion); Younger Child BOY (Gareth)

Outcome #2 – Older Child BOY (Hyperion); Younger Child GIRL (Swan)

Outcome #3 – Older Child GIRL (Swan); Younger Child BOY (Hyperion) NO LONGER POSSIBLE

Outcome #4 – Older Child GIRL (Swan); Younger Child GIRL (Moiraine) NO LONGER POSSIBLE


As you can see, Outcome #3 and Outcome #4 are no longer possible, because we know the OLDER child is a BOY. Therefore, we only have two outcomes to look at, and those two give us equal chance for a BOY or GIRL.

Both Situation A and Situation B are closed loops. This means that the possible Trials have already taken place. By looking at those Trials, and then adding the discrete information we are given we can determine the answer. What makes the probability of Situation A 2/3 and Situation B ½ is determined but what Outcome Trials you are examining once your new information is added. Does that make sense?


The other main difficulty in determining probability is figuring out what knowledge if useful. Who has the information and when do they have it? Another hypothetical problem using the same information will help us see when the loop is closed and when it is open. (Actually, it is a closed loop of a different type, but I am trying to make it easier to understand, so bear with me. Get it? Bear with me?)


Bear's Bastards - Situation C

I am in the hospital and Bear comes to visit me, bringing his two kids I did not even know he had. The first kid through the door is a BOY named Hyperion. What is the probability that the second child is a GIRL?

Answer - ½.

Some of you will recognize that in essence, this is Situation B in different clothes, but it may help to look at it as an Open Loop. In other words, I have zero information on Bear having kids and what gender they might be. The first kid's arrival did not change that, as the second kid could be either boy or girl. In effect, because the situation happens live the probability has "reset." Many people instinctively think this way in all probability situations. They rightly assume that probability has nothing to do with past results. If I flip a coin 10 times in a row and all ten times it comes up heads, there is still a 50/50 chance the next time will be tails.

But because in Situation A we were looking at a finite set of Outcomes with limited information, we were able to see the probability shift.


To understand the value of "outside knowledge" more fully, we turn to a different problem. On the surface, it will look a lot like yesterday's Let's Make a Deal, but we shall see it is very different in result.


Bears' Mating Habits - the Pop Quiz

You come into class one day only to be given a pop quiz by your teacher on the mating habits of bears. The quiz is multiple choice, with three choices for each one: A, B and C. Needless to say, you are unprepared, and start randomly guessing.

1)A
2)B
3)C
4)C
5)B
6)A
7)A
8)B
9)C

The Teacher sitting at her desk starts to feel sorry for the class and wants to help. She calls out, "Class, the answer to #7 is not C."

The question is: do you switch from A to B?

From yesterday's probability problem it might seem like you would switch. But in this case, it does not matter at all, since the probability is 50/50 with either guess. How can this be?

The difference from yesterday is that the Teacher is sitting up at her desk and has no idea what each student has answered. Trying to be helpful, she just calls out that C is not correct for #7. The only way this information could have helped you is if you had picked C! In that case, you would definitely want to switch. However, switching from A to B is not going to help you any, at least probability-wise.

In yesterday's example, I knew what door you picked, and I know what doors DID NOT hold the billions of dollars. I showed you one of those doors, which I was able to do regardless of whether you would picked the correct door (since two of them led to Hell). Because of this knowledge, it kept your guess in a Closed Loop, and had an effect on the probability, unlike the pop quiz where the teacher's comment does you no good. Capische?


Well, I hope this helps you out. If you do not understand it feel free to write me at hyperioninstitute@gmail.com and we can go over it some more. If you do understand it, feel free to start winning bets at dinner parties and the like. Just make sure I get my cut.

(Now what are the probabilities of that?)


Lurking behind Door #3 with Bear's Bastards,


Hyperion
August 22, 2007

9 comments:

C3P0 said...

Surrender is a
perfectly acceptable alternative in extreme circumstances.

Bear said...

Everything is perfectly fine here...

Except for Hyperion's claim that the knowledge that he had when revealing the fruit was the same as the knowledge Monty Hall

has when he reveals a door. This is simply not true. I fear that Hype is distracted by dreams of my sister again. Let me

explain (and don't groan!):

In many ways the fruit packet situation is very similar to the door situation. In each situation there are three items

(doors and fruit packets). Each item has two states (Hell/Prize and peach/pineapple). The first fruit packet falling from

the bed this is analagous to the contestant choosing a door. It's a random selection of one of the three items.

At this point, however, the situations diverge. This is the point where the second item is revealed (door is opened

Hyperion is correct when he describes Monty Hall and says that Monty knows which door contains the right answer and which

door was initially picked. Thus, this information guides Monty to intentionally never reveal the winning door. Monty does

not randomly open a door. If he did, he could have picked the winning door (provided the contestant does not choose it

first). But he NEVER reveals the winning door. It's not a possible option given Monty's behavior.

On the other hand, when Hyperion reveals the second fruit packet, he knows which packet was initially picked, but--unlike

Monty--he does not know which packet is the pineapple packet. Thus, Hyperion's behavior was not guided by anything other

than random selection when he revealed the second packet. It was certainly possible that Hyperion could have picked the

pineapple packet.

To summarize the difference:

Monty knows the correct door and thus never opens it, eliminating possible outcomes.
Hyperion does not know the pineapple packet and thus could definitely eat it first, so he does not eliminate possible

outcomes.

In this case the intuitive answer is the correct one, so people, don't fret. You are smarter than Hyperion gives you credit

for.

In fact, you can perform a very simple experiment to convince yourself: Just get three coins and set them on the table. Make

sure two of them are heads and one of them is tails. Next, randomly shuffle them around so that you don't know which is

which (cover them up by cups or pieces of paper). Next, randomly set one of the cups to the side without looking at the coin

underneath (equivalent to the random packet falling). From the remaining two cups, randomly select one of them and look at

it (equivalent to hyperion seeing the peach packet and eating it). If it's the tails coin--equivalent to the pineapple

packet--you have to start the experiment over, because need to match what happened to Hyperion. In other words, in order for

the experiment to be properly set up, you need to randomly select a heads coin. Once you've done that, you've got a heads

coin revealed, and two other coins hidden. Now, the experiment: See if the tails coin is the one that you set aside--or,

equivalently, the pineapple is the packet that is on the floor.

If you perform this experiment 100 times, I guarantee you that the tails coin will be the one set aside 1/2 of the time

(about 50 times). But guess what? One of my co-workers and I wrote a program that did the trial 1,000,000 times (1 million).

Out of all of those trials, a peach was randomly eaten 666828 times (66.6828%), as expected, and out of the number of times

a peach was randomly eaten, the pineapple was on the floor 333259 times (49.97676%). My friends, that is 1 out of 2. If you

want, I can send you the program and the source code.

I challenge Hyperion to perform this experiment himself. I also challenge him to explain the difference that I have laid out

between the two situations. I also challenge him to examine the list of possible outcomes, as I did yesterday (and as he did

today with the bear cubs), and explain how the probability is 2/3. The answer, my friends, is that it can't be done.

Koz said...

Bravo Bear. If there is a way to beat Hyperion it is with statistics - the numbers don't lie.

I actually wrote a similar program but since you went into so much detail I feel no need to share my results here.

Hyperion said...

Bear, the very fact that you ran a simulation tells me that either you're not understanding what I said or I communicated ineffectively or both. You don't need to run a simulation to understand the problem. I removed the peach choice knowingly and offered the past Hyperion the choice. It's the exact same situation, I may have just worded it poorly, which I have now fixed. Go back and read it again and see that by putting one word in I omitted before the whole thing makes more sense. You need run no more tests than you would in Monty Hall.

And Koz, be wary of numbers. Often they are used to prove things, but often the real question is what the issue under discussion really is.

Let this be a lesson to all of us. Bear is smarter than you are, probably smarter than anyone you know. He actually wrote a program to test all this out. That's cool. However, because of a miscommunication, it tested the wrong thing, which led to a misunderstanding. You remember that the next time your favorite politician uses a number to "prove" whatever it is he/she/mitt wants to do to us.

Bear said...

OK. It seems that none of my challenges were attempted.

Your original question was

What were the odds that the pineapple package WAS NOT the one that fell to the floor, and was still on the bed with me?

This clearly asks what is the probability of the peach being on the floor after everything had transpired. It mentions nothing about "past Hyperion". But in the interest of the discussion, let's see what would happen for past Hyperion.

Situation 1:

Suppose that you tell past Hyperion "The first packet on the bed is a peach packet." Then here are the possible outcomes:

On Floor -> 1st On Bed -> 2nd On Bed

Peach A -> Peach B -> Pineapple
Peach B -> Peach A -> Pineapple
Pineapple -> Peach A -> Peach B
Pineapple -> Peach B -> Peach A

These are the only 4 possible outcomes from the given information. The pineapple packet can never be the 1st packet on the bed. The pineapple is on the floor in 2 of these outcomes. Hence, a 2/4 (or 1/2) chance of the pineapple being on the floor. Obviously, this is the same situation as for the "future Hyperion" because once past Hyperion knows what the first packet on the bed is he has the same knowledge as future Hyperion.

Situation 2:

Now suppose that you tell past Hyperion "One of the packets on the bed is a peach packet." Then the possible outcomes here are:

On Floor -> 1st On Bed -> 2nd On Bed

Peach A -> Peach B -> Pineapple
Peach A -> Pineapple -> Peach B
Peach B -> Peach A -> Pineapple
Peach B -> Pineapple -> Peach A
Pineapple -> Peach A -> Peach B
Pineapple -> Peach B -> Peach A

Note now that there are 6 possible outcomes. Why? Because you did not tell past Hyperion which packet on the bed was a peach packet. Is it the 1st or the 2nd one? You have given him useless information because he already knows that one of the packets on the bed has to be peach (i.e. they can't both be pineapple since there is only one pineapple)! The chances of the pineapple being on the floor in this situation are indeed 2/6 (or 1/3). But, this is not the same situation as the one you described because past Hyperion does not know the same information as future Hyperion. Future Hyperion knows which packet on the bed is a peach packet, whereas past Hyperion does not. In that case if I were past Hyperion I would definitely take the two on the bed, if I wanted to increase my odds of getting the pineapple.

Future Hyperion, though, is stuck with 1/2 odds of the pineapple being on the floor. If his chances were 2/3, then my experiment would have shown that. However, it showed 1/2. The experiment is pretty simple. Do the experiment, and see how many times the pineapple is on the bed. Experiments prove that it's not just theoretical nonsense that people are talking about. It shows a real world application, and convinces you of the validity of the theory. I don't have any underlying motives here. I'm not trying to buy votes, or push any agenda. It is what it is.

Hyperion said...

At some point we're going to look back on this episode as a Monteclair moment. It shows, if nothing else, that science and math, while endlessly useful and a necessary part of life, sometimes misses the point. It also shows the danger of being loose in speech. My original example was always meant as an intro to Monty Hall, and not told exactly. I now wish I'd left it out, since it caused so much confusion. Sigh.

I swear to all that is holy on a stack of bibles that no experiments need to be done on this. All of your experiments are good and impressive, and some have even caused people's eyes to explode, which is always good, but no experiment needs to be run.

Now, before you tell me I'm wrong. Consider this: I am able to say these things because I am the one who created the situation. Much like when I write a story and I tell someone what the character means. They may not like it and they may have gotten something else, but having created the character, I most assuredly know what he means.

To iterate, the situation with the pineapples and Monty Hall are exactly the same. If I did not explain it right the first time, I am sorry. As I explained in this column, I left information out of the first example because I was trying to quickly get through it and get to the more entertaining (and easy to understand) version of Monty Hall.

The situations ARE the same, and anyone not seeing that has either missed it, or I have communicated poorly. Again: I freely admit to that. This is why I wrote a second column, trying to make things more clear.

However, it seems that my telling you they are the same and attributing misunderstanding to initial unclarity is not enough, so let me try this a different way.

Hyperion came on Monty Hall to play Let's Make a Deal. There were three doors. Behind the doors were three bags of Crispy Fruit, two peach and one pineapple. Hyperion wanted to pick the pineapple one, because if he did so he got to have an illicit affair with Bob Dole.

Hyperion selected door #1. Before it could be opened, however, Monty opened Door #3 and pulled out the Peach fruit and ate it. Monty did this knowingly to confuse Hyperion. Monty then asked Hyperion if he would like to switch doors before it was revealed which one contained the pineapple. Having recently been schooled in probability, Hyperion said yes.

That was the situation. All the tables and logic are brilliantly derived, but are in this case not valid because they describe a different scenario.

I think where people might get bogged down is in assuming that the peach was removed randomly, and as such the probability resets. THIS IS NOT THE CASE.

This is what I was talking about today. Future Hyperion had full knowledge. He then offered past Hyperion the choice, which stayed a valid 2/3 for switching, because past Hyperion was in a closed loop. he had no further knowledge. In essence I was retrofitting the Monty Hall problem to fit my scenario. How could I do that? Because I made it up. It was an illustration.

Look, what the question actually requires is to morph the thinking about two different Hyperions in different times and different places. This is difficult for some who are taking everything literally. And that's not wrong. What I learned from this is to not be so colloquial with my speech. I created the scenario artificially in my head, planning on writing about Monty Hall. What perhaps I should have done is just start with them.

BTW: the pineapple was on the floor.

Bear said...

I understand that you think the two scenarios are the same. However, when you fit your packet scenario to include Monty Hall you are making a fundamental change in the situation. I don't know how much simpler to explain the difference, so let's answer the following questions:

Q: When you ate the peach on the bed, did you know where the
pineapple was?

A: NO. Therefore the peach was removed randomly.

Q: When Monty eats the peach, does he know where the pineapple is?

A: YES. Therefore the peach was not removed randomly.

That is quite clearly a huge difference, so the situations are not the same.

----------

So far I have:

1) Clearly explained a fundamental difference between the two situations, which gives reason to question whether or not they really are the same thing.

2) Laid out all possible outcomes and shown how the number of outcomes where the packet is on the floor divided by the total number of possible outcomes yields 1/2. This is the exact same method that you used in your analyses of the bear cubs yesterday.

3) Performed an experiment that validated my theoretical conclusion.

None of these points have been directly addressed, except for statements saying that they are wrong or unnecessary without direct explanation of why they are wrong or unnecessary.

i) Do you not agree that there is a difference? If not, why are they the same thing? If they were the same thing, then you would have had the same knowledge as Monty when you both eat the fruit. Do you think that you did have the same knowledge? How so?

ii) If your tabular method for determining probabilities yesterday for the bear cubs was valid, why do you invalidate the method when I use it? Why is my use of the method invalid? There must be a specific assumption that I am making when I lay out my tables that you believe is incorrect.

iii) You're right, I could be spouting off numbers just to make my argument look better. In a sense I am, because I believe that they support my argument. But I'm not just pulling numbers out of my ass. The whole point of the experiment is that anyone can do it and see for themselves. You say my experiment is testing a different situation than the one you initially described, but how is it different?

Hyperion said...

I begin to see how the Palestinians and Jews aren't able to work things out. If two really good friends cannot connect and are like ships passing in the night, what hope do mortal enemies have? (Then again, if we keep going we might be dueling by September.)

The real problem might be the media of discussion. I have a feeling that one five minute phone call would make everything clearer. However, in the furtherance of science....

I'm not trying to disrespect all your hard work. I'm not trying to ignore it. I'm not even saying you're wrong. I'm just saying you're talking about something else.

The situation with the peach and the pineapple didn't actually happen. At least, not in the way I described. It was retrofitted to segue into Monty Hall. In the real story I did not pick the peach at random and I knew where all the ones were.

When I was giving you assurances that the situations were the same, I was not trying to just say I'm right because I'm right. I was trying to say that it would be literally impossible for the situations not to be the same.

Why? Because I made up the pineapple/peach situation specifically to be like Monty Hall. I retrofitted all the important data to be the same situation.

The only possible criticism, on that has been fairly leveled and I have acknowledged several times, is that in my attempt to tell a "folksy" story I left out details and rushed others, leading those who are on the ball like yourself to see areas where the information may have looked wrong. I went back and fixed it, but you are left with your initial impression. That's valid from your standpoint, but it doesn't change the fact that they are the same situation, if only because I created them to be that way.

Look at it this way: let us say that I was describing George Bush to you for you to draw. However, you and I interpret things I'm saying differently, and you end up drawing Jeff Foxworthy from the description I gave you.

You make the case that from what you heard and understood to be the case I was describing Jeff Foxworthy. But regardless of that statement's validity from your perspective, the fact remains that in my head I saw the real George Bush and attempted to describe him, however it turned out.

Does that make sense? I was always describing the Monty Hall situation, and if you feel I did a poor job, well, then hopefully I addressed that with the secondary column. But I can't have been describing any other scenario, because that wasn't what I set out to do. All I could have done was poorly do the task at hand.

Which clearly I seem to have done.

Bear said...

I understand now. The point where I was misunderstanding you was when you said that because you saw the peach packet you didn't choose it at random. But you did not mention that you knew where the other packets were as well. In that case, I agree that it is the same situation. I had a feeling something was up. Nevertheless, it's been fun, if only by causing aneurisms to people in several states and a few countries.