# Statistical question about having 2 boys if one is born on Tuesday

Some of you may have come across this problem before:
If I have 2 children, one of which is a boy, what are the chances the other one is a boy?
Most people would respond with 50%. The answer is actually 33%. I have no issue with this answer. Initially there are 4 possible combinations, GG, BG, GB, BB. You know I have one boy so you eliminate GG from the group and 1 out of the 3 left over is BB.
However, if I tell you that I have 2 children, one of which is a boy born on a Tuesday, the chances of me having 2 boys now change to 13/27 or about 48%:dubious:

If you want an explanation follow http://www.maa.org/devlin/devlin_04_10.html

I can follow this explanation to its conclusion and I have even made a spreadsheet with 50000 trials to confirm both ratios. When I only look at gender, the ratio is around 33%. When I look at gender and day born, the ratio is 48%.

The thing that bugs me is my boy from the first example would have been born on one of the 7 days and the logic works no matter which day I use.
So if I say to you I have 2 children, one of which is a boy and ask what the chances of the other being a boy are, the answer is 33% However, if I ten tell you what day of the week he was born on, all of a sudden the odds have somehow changed. :smack: Is there something I am missing.

I sort of have a problem with the original premise.

Yes, you can eliminate GG, but since we you’ve given me no information about order, and it appears to be an irrelevant issue, then BG and GB are the same thing. Thus leaving us with either BB, or BG/GB meaning that there is a 50% chance of the other being B or G.

If order of children is an issue, and you told me that your first child was a boy, it would still be 50% because then we could eliminate GG and GB from the mix, leaving us with BG and BB or a 50% chance of the second one being a boy.

What am I missing?

Think of it this way: the more specific the conditions you place in the “one of my children is X” statement, the less likely it is that you would have two children satisfying that condition. So the choices are likely to be between “BX” and “GX”, not “XX”.

The trick with this question is that it matters how you get the information. The answers you give there pretty much only apply if you say you have two children, and then someone asks you “is at least one of them a boy who was born on Tuesday?”, and you say “yes”. But this situation (or situations equivalent to it) is extremely contrived, and hardly ever comes up in real life, which is why the answers seem so counterintuitive (because our intuitions develop to deal with the sorts of situations which really do come up often).

No, it doesn’t, because for most days, your answer to that question would have been “no”, in which case we get a completely different piece of information.

Here’s the best I can come up with:

If all you know is that someone has 2 children, there are all sorts of possibilities for what those children might be: all sorts of combinations of sex, day-of-week-of-birth, month-of-year-of-birth, eye color, handedness, etc., etc.

If you’re told that at least one of the children is a boy, that eliminates 25% of the possibilities, leaving you with the other 75% which are evenly split among the BG, GB, and BB possibilities.

If you are also given some additional information, this allows you to eliminate more of the remaining possibilities. But those possibilities don’t necessarily get eliminated in equal proportions from the BG, GB, and BB blocks.

If the additional information is that at least one of the children is a boy born on a Tuesday, this eliminates 6/7 of the possibilities in the BG block, 6/7 of the possibilities in the GB block, but siginifcantly fewer than 6/7 of the possibilities in the BB block, since, with two boys, there are more ways of getting at least one boy-born-on-a-Tuesday.

We’ve have seen the “other child” question discussed to death, and the conclusion I’ve reached is that it’s not a well-formed problem.

The true assignment of a probability requires an “ensemble”, a set of identical situations or repeatable experiments; the probability of A is just case-A divided by total-instances. In a well-formed probability question the ensemble or experiment to be repeated (or that hypothetically can be repeated) is clear or can be reasonably inferred: flip a fair coin; all twelve-year-olds in Houston; all women who have given birth to a son; …

The difficulty with the other child problem is that the ensemble is ambiguous. Starting with two-child families is easy. When you now “know” one child is a boy, you’ve selected a new ensemble, a sub-set of the original - but the manner in which you “know” this is not specified and therefor it is not clear which subset of families you are actually choosing. I can think of perfectly reasonable ways of selecting at-least-one-boy families that would give either the 50% answer or the 33% answer.

It would be the same for the boy-on-Tuesday families. Do you first select out families without a boy? Do you select children born on Tuesdays and then eliminate the girls? Do you weed out families without Tuesday birthdays and then cut out those with two girls? The answer you get will depend on how you know there is a Tuesday boy.

It often isn’t well-formed, but it certainly can be. Also, even with a well-formed problem, many people seem to have a problem with it.

I’m not quite sure exactly what you mean by the first two cases above, but in the third case, you’ve only assured they have a boy and have a child born on Tuesday, not that they have a boy born on Tuesday.

For the OP: I find it easiest to make a grid. In the usual problem, there’s a 2x2 grid, and you eliminate one of the possibilities. In the one with the day-of-week as well, make a 14x14 grid with rows and columns labelled BSun, BMon, BTue, … BSat, GSun, …, GSat , then eliminate all but the column and row with BTue. That leaves 27 possibilities, and you can work out the probabilities from that.

BG and GB are different because they imply some kind of ordering. It could be age, height, darkness of hair, anything. Anything that makes them distinguishable, such as your later example of whether the child is the first child or not.

The original wording of the puzzle is ambiguous, although it disguises its ambiguity quite cleverly. Recall the wording: “One of them is a boy. What is the probability that the other is a boy?” Who is this “other”? It sounds like it refers to a specific, identifiable individual. But it doesn’t, because the thing it is being contrasted with, “one of them”, does not identify an individual. It refers to either of them.

Thanks to everyone who has posted. As I said, I understand the explantion of why the odds change. I can arrive at 13/27. I think Chronos gets what my problem is, but I don’t understand his response. Imagine this:
I take you on a trip to some random location to meet some random 2 child family. We are standing at the door and one of the children walks out. He is a boy. I then say “There is a 33% chance that the other is a boy”. Then the boy says “I was born on a Tuesday”. Should I then say “Actually make that 48% chance”?

No, because in that instance you know which child is a boy. Therefore, the probability that the other child is a boy is 50%.

Expanding on ultrafilter’s comment, as soon as you have a particular child specified, there can no longer be uncertainty about which is “the other”.

And going back to the original question, the phrasing, “I then tell you what day of the week he was born on” suggests that you at least have a particular child in mind you’re telling me about. But who exactly is that?

I accept the ordering, so of GG, GB, BG, BB the GG option is out, leaving us with GB, BG and BB. We know you have one boy, but we don’t know whether the boy is the first or second child. So, there is a 50% chance of either based on what we know. So, the way I read it we have to divide the likelihood of either GB or BG by 50%.

So to me the odds (bolded is the current boy that we know about) are:

BG 25%
GB 25%
BB 25%
BB 25%

Thus leaving us with a 50% chance that the non-bolded child that we don’t know about is a G.

It’s not 50%. We know that at least one of them is a boy, but when you say “the” boy, it suggests that only one of them is a boy, which would make the boy identifiable.

It depends on whether we randomly select a boy (out of all boys who have exactly one sibling), or whether we randomly select a pair of siblings (out of all pairs of children with at least one boy). Your interpretation is the former way.

It depends on the sample space. Assuming G and B are equally likely and independent,
(1) Out of all boys who have exactly one sibling, in 1/2 of these cases the sibling is a boy.
(2) Out of all pairs of siblings in which at least one is a boy, in 1/3 of these cases the other is also a boy (i.e. both are boys).

No, I take that into account above with the two potential outcomes of

BB and BB

Those together account for the possibility of both are boys (and add up to 50%).

Got it. Thanks.

Seems to me that the question is intentionally ambiguously stated to have an obvious and nonobvious defensible answer.

Those outcomes are not possible if only one of them is a boy.

The problem with assigning a probability is that you’ve described a single event - and single events don’t have probabilities. (What’s the probability that Africa split from South America?)

You have to imagine extending this to a repeatable process. For instance: Say you visit many families; ignore those with fewer or more than two children; ignore those in which the first child you spot is a girl; ignore those in which the first child you spot was not born on a Tuesday. Now, of those remaining families, note the sex of the unseen child. It will be 50% boys, 50% girls.

You can imagine different Tuesday-boy “experiments” which will have different selection processes and will give you different percentages.

Says who?

I’m going to again suggest making a 14x14 grid like I described above; it has 196 possibilities.

Suppose instead of seeing a child, one of the parents says “We have at least one boy.” At this point you can rule out all 49 of the two-girl possibilities, and you’re left with 147 possibilities. 49 of those are two-boy possibilities, and the other 98 are one boy and one girl.

If instead the parent says “I have a a boy born on a Tuesday”, you can rule out all but 27 of the possibilities. The difference in probability between the two cases comes from having a different set of possibilities in each case.