The Hyperion Chronicles

[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