h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

This is an interesting example of permutations, perhaps; but it does not represent probabilities. It's actually a better example of sophistry.

When the question is asked about the gender of the two children, and one knows that one of the children is a boy, that child is no longer in question and doesn't fit into probability.

While it's true that the GG combination can no longer be true, there is nothing about the gender of the known boy-child that has any bearing on whether the second is boy or girl, given the 50:50 ratio mentioned.

The only way the gender of the known boy-child could have an impact is if the two were monozygotic twins.

It ain't a paradox, alright. It's a slightly counterintuitive probability question.

If the older child is a boy, the options are BB and BG, 1:1.

If one child is a boy, the options are BB, BG, and GB, 1:1:1. A two-thirds chance that the other child is a girl.

TRiG.

Thanks Trig! But I already understood the Truth Table. I just submit that it doesn't apply in this case.

What the table actually says is that there are two combinations that include the B and only one that includes the G.

What I'm suggesting is that knowing the gender of one has no bearing on the gender of the second. The probability that a couple's second child would be a B or G, in a 50:50 probability world, is equal.

Insisting otherwise is like saying that if you've flipped heads twenty times, with an un-rigged coin, that you could calculate the 'probability' of the next coin coming up heads or tails. You simply can't; each flip is going to be fifty-fifty.

In the situation described in the entry, each is a separate probability, with no dependency against past or future flips.

Think about it and let me know if you still think I'm wrong. I'm certainly capable of making an error!

Hi, just stumbled on this.

I think the error that the article is making is: first it says that it doesn't matter whether or not the known boy is older or younger, but then then insists that BG and GB should be treated as separate outcomes. If you are goung to allow BG & GB, you should also allow BB & BB (oldest/youngest & youngest/oldest) as separate outcomes. That puts the probability back to 50:50.

However, re-reading the article, it doesn't say that there is a known boy, simply that one of the children *is* a boy, in which case the article is right.

If you take 400 two-child families there will be 100 BB, 100 GG, and 200 BG if you ignore age. What the question does is take away the 100 GG families, and it's that act of selection that skews the probabilities.

Well, it does say "states that at least one of the children is a boy". To me, that means a known boy child.

[I think the error that the article is making is: first it says that it doesn't matter whether or not the known boy is older or younger, but then then insists that BG and GB should be treated as separate outcomes. If you are goung to allow BG & GB, you should also allow BB & BB (oldest/youngest & youngest/oldest) as separate outcomes. That puts the probability back to 50:50.]

I absolutely agree that it's 50:50, but not for by reason of correcting the truth table.

[If you take 400 two-child families there will be 100 BB, 100 GG, and 200 BG if you ignore age. What the question does is take away the 100 GG families, and it's that act of selection that skews the probabilities.]

Given the way you corrected the truth table, I'm sure you meant:
100 BB, 100 GG, 100 GB, and 100 BG.

But, in any one of those 400 cases, the gender of the first-born child bears no relationship to the probability of the second child's gender.

Now, the Three Door Puzzle is a much better example of increasing one's odds of the predictive powers of probability, because it's all one set of possible outcomes, with each choice changing the probability of the next. Birth doesn't work that way. Like I said, it's precisely the same as two coin tosses, at least nine months apart.

Here's another way of thinking about it. The fact that the children are related may be causing some confusion.

I tell you that two people, and two people only, entered a closed conference room. I mention that a male was the first to go in.

Now try to predict the gender of the second person!

50:50

But the question is talking about families that already exist, and in the population as a whole there are twice as many families with one of each as there are families with two boys. The thing that skews the probabilities is saying "one of the children is a boy". You've made a non-random selection, in exactly the same way as Monty Hall getting rid of one of the goats.

Another way of looking at it - if you filled a big room with boys who have one sibling, picked a boy at random and said "Your sibling is a girl", you'd be right two times out of three (on average).

I guess that, in a world filled equally with boys and girls, say 3,000,000,000 of each, it would still hold true. If you take away one by declaring their gender, you have marginally reduced the population of that gender and increased the chances that the next one will be the alternate gender.

In reality, there are additional factors to consider. In my example of the room, it would skew the probabilities if you knew what was in the room and/or the reason for their entering it.

However, in the arena of real-life siblings, the 50:50 factor is not always based on chance. High altitude pilots and deep sea workers have a much higher incidence of female children.

I know more families with all boys or all girls than families with equal numbers. I suspect that having an equal number of each is far less likely than having an uneven number. All sorts of reasons, from the woman's Ph factor to spermatoza motility, can skew the results far beyond the probabilities; even if, at the end of the day, the numbers even out.

Then I take into account that I'm much more likely to meet up with one of my brothers for some activity than with one of my sisters.

I wonder if it would still make sense if we knew that there were another scenario that offers the same probability matrix, say the probability that if you reached into a closet with several pairs of shoes, and pulled out a right shoe. I would think that the probability of pulling out a left shoe on your next random grab would be slightly higher.

The problem with that scenario is that nearly 100% of shoe buyers get one of each with every purchase. Not so with families.

I don't see that working at the very small numbers involved in siblings. It's much more a coin-toss scenario, not subject to probabilities, just permutations. I just don't buy that the coin tosses of genetics are affected by previous tosses.

I've been trying to understand why the concept doesn't work for me, and I think I know from where my objections are coming.

In statistics, we calculate probabilities all the time; but we understand that small datasets do not follow the rules. If we don't have a minimum of 30 data points, our confidence in our calculations starts quickly going down.

If we were talking a dataset of 30 families, each of which have an even boys:girls ratio, I'd start believing that the odds would work for me better than a casino. It's just that the dataset is too small and undefined. We don't even know that there is a sibling of the other gender, and it still appears that the sibling could equally be of either.

I think post 9 gets it.

Let's look at the sample: families with two sexed children. (Families with more or less than two children, and families with intergender children are not part of the sample.)

Four possibilities: (a) BB, (b) BG, (c) GB, (d) GG. Each possibility is, I submit, equally likely, in our theoretically statistically perfect world. (Does anyone disagree?)

The first question eliminates options c and d from the sample. The probability of option b turning up is, as the article says, one half.

The second question eliminates option d from the sample. The options a, b, and c still have the same relative weight, so the possibility of any one of them turning up is one third. The probability of b and c, taken together, is two thirds, as the article states.

TRiG.

I fear that we are not always using equal terms. Probability and Possibility are two different things.

When we talk about how many possibilities, we're talking permutations; which combinations are possible. When we switch to Probabilities, what is the chance that of each possibility to occur.

You've actually described that pretty well; there are four possible combinations (outcomes) and you've eliminated those that don't fit the givens of the problem.

So, what if we replace the kiddies with coin tosses. Shouldn't the outcome be the same? If you flip a coin and it comes up heads (the oldest flip) what is the probability that the second flip (the youngest flip) will be the opposite?

If you flip a coin twice, and the first flip comes up heads, what is the probability that the second flip is tails?
If you flip a coin twice, and at least one of the flips is heads, what is the probability that the family other flip is tails?

Four possibilities: (a) HH, (b) HT, (c) TH, (d) TT

I submit that each time you flip the coin you have an independent outcome. I also submit that in a 50:50 world of children, the same holds true for siblings.

And, if you discover a room that purports to contain a randomly selected family that happens to have exactly two children, you'll admit that the probability of them having any of the four combinations is equal. So, if you discover the gender of one, how could that ave any effect on the remaining child.

If the parents called out "First-born, come out please", and a boy appears (his name is Schrodinger), what does that tell you about the gender of the other child?

Question 2:
The extensional definition of outcomes {BB, BG, GB, GG} - in which B = boy, G = girl and the first in each pair is the eldest - doesn't match the constraints in question 2. For question 2 the extensional definition of outcomes would look like this: {BB, BG} - where B = boy and G = girl. Each event has a probability of 1/2.

Note: GG is not included because at least one of them must be a boy.
GB is not included because age is not a factor, therefore GB is mathematically equivalent to BG and therefore only one of them should appear as a valid outcome.

Thus the probability of the 2nd child being a boy is 1/2.

Another perspective:
You can break question 2 down into two equally likely versions of question 1. The known boy must be either the eldest or the youngest.
Split the problem in two: in one scenario the known boy is the eldest and in the other scenario he's the youngest. This gives two equally likely (probability 1/2) sets of outcomes: {Bb, Bg} and {bB, gB} - where B or b = boy, g = girl, the eldest is first in each pair and the known child is capitalized.
Combine the two halves again: {Bb, bB, Bg, gB}. Listing the outcomes in this way helps disambiguate the problem by describing 4 equally likely outcomes, each with a probability 1/4.

Two of these outcomes {Bb, bB} show the unknown child is a boy (1/4 + 1/4 = 1/2) and the other two outcomes {Bg, gB} show the unknown child is a girl (1/4 + 1/4 = 1/2).
Thus the probability of the 2nd child being a boy is 1/2.

The apparent paradox is a fallacy that comes from applying an incorrect model. It's incorrect to list BG and GB as two separate outcomes in question 2's model. Without age as a factor, BG and GB are just two ways of describing a single outcome. The extensional definition of outcomes is
{BB, BG}
Only if you split the problem into two (as above) and then infer information about the age does it become appropriate to list BG and GB separately; at which stage, the extensional definition of outcomes must list BB twice. As a more concrete example, if you make a distinction between {Sally Frank, Frank Sally}, then you must apply that distinction universally, hence {Frank John, John Frank}.
{BB, BB, BG, GB}
You can't mix and match the two paradigms, as has been done elsewhere, without introducing an apparent paradox / false dichotomy.
{BB, GB, BG} This is incorrect, it does not apply age as a factor universally. GB and BG are the same event, there is no significance to the order in which it is written. We must pick one of them and only one of them.

Did you just say the same as me? The frilly brackets distracted me.