## The Tuesday Birthday Problem 981

Posted
by
kdawson

from the if-it's-tuesday-it-must-be-a-girl dept.

from the if-it's-tuesday-it-must-be-a-girl dept.

An anonymous reader sends in a mathematical puzzle introduced at the recent Gathering 4 Gardner, a convention of mathematicians, magicians, and puzzle enthusiasts held biannually in Atlanta. The Tuesday Birthday Problem is simply stated, but tends to mislead both intuitive and mathematically informed guesses.

*"I have two children, one of whom is a boy born on a Tuesday. What's the probability that my other child is a boy?"*The submitter adds, "Believe it or not, the Tuesday thing is relevant. Well, sort of. It's ambiguous."
## Re:Rubbish (Score:3, Interesting)

## Re:Let's try it without reading TFA (Score:5, Interesting)

It's playing games with words and attaching significance to two sets that in any practical case I can think of would be considered one.

The argument is that if you were to consider it as a set, there are four possible ways for your children to be distributed:

1. (Boy, Boy)

2. (Boy, Girl)

3. (Girl, Boy)

4. (Girl, Girl)

We already know that your children can't possibly fall into the fourth set, and so looking at the sets it appears that the probability should be 1/3. But this misses one minor point - you've added an extra set which only makes sense if you wish to attach significance to the order in which the children were born (Sets 2 and 3). But as soon as you do attach that significance, the information you are given in order to establish the probability of any particular outcome (eg. the boy is older) allows you to eliminate two sets rather than just one.

## Re:Probability (Score:3, Interesting)

There is a very small chance it will land balanced perfectly on it's side

Has anyone ever seen that happen?

Yes, I've had that happen once with a 10Fr coin (very similar in shape to the current 1 euro coin). The ground was irregular which probably helped a lot.

## Re:Well? (Score:2, Interesting)

Indeed, is everyone drunk or something...

First, The question doesn't say the other (this does not mean older or younger...) child was not born on a Tuesday, maybe the questioner meant to include this info but they failed to.

Second, the probability that the other child is a boy is either 1 or 0, it's something that has already occurred... The questioner probably meant to ask "What is the probability that if you guess the other one is a boy you will be correct?"

So if we correct the question to read as it was likely intended to be read:

"I have two children, one of whom is a boy born on a Tuesday. The other child is not a boy born on a Tuesday. What is the probability that you will be correct if you guess that my other child is a boy?"

So you think to yourself, well assuming boys pop out of that mom just as easily of girls and doesnt prefer to do it any particular day of the week, that means its 1/7 chance it might have happened on any given day and the likelihood it was a boy is 50:50 for 6 days of the week and then 0 for tuesday. So you multiply .5 by 6, add zero, and take the average (divide by 7) to get 3/7.

The real confusion occurs due to the use of odd numbers... Imagine a world where everything was found in sets of twos, people had 2 heads, 4 arms, etc. They would always be dealing with eating animals that were siamese, if they wanted to hunt by throwing rocks or whatever each siamese would throw a rock so they would use two rocks. In this world I would say that what we call the number 2 would actually be like their number 1, and what we use as unity, or one, would be for the siamese called a half. Therefore their numberline would go 0, .5, 2, 2.5, 4, 4.5, 6, 6.5, 8, etc.

This is actually more reflective of reality in that, deep down, math and counting are extensions of logic, and the fundamental unit of logic is a true-false statement which is basically a set of 2. True is only 1/2 of the total possibilities for any given logical statement. For example say you have counted one rock, what that actually represents is both having one rock in your presence butt also, concurrently, not having counted other than one rock, so in essence you have counted two different things and are representing them with a number supposed to correspond with one thing. Wouldnt it make more sense to just use "two" to represent the one thing youve counted?

The probability of guessing correctly by saying the second child is a boy would therefore be 1/2(6), or 3, divided by 6 and a half, which gives you 6 out of 12 and 1/2 odds.

## Re:Ordering and Convergence (Score:2, Interesting)

The comment you quoted is incorrect. There are 28 combinations of boy/girl and day of the week, but only 27

uniquecombinations. Here they are:1 Boy Thu, Girl Sun2 Boy Thu, Girl Mon

3 Boy Thu, Girl Tue

4 Boy Thu, Girl Wed

5 Boy Thu, Girl Thu

6 Boy Thu, Girl Fri

7 Boy Thu, Girl Sat

8 Boy Thu, Boy Sun

9 Boy Thu, Boy Mon

10 Boy Thu, Boy Tue

11 Boy Thu, Boy Wed

12* Boy Thu, Boy Thu

13 Boy Thu, Boy Fri

14 Boy Thu, Boy Sat

15 Girl Sun, Boy Thu

16 Girl Mon, Boy Thu

17 Girl Tue, Boy Thu

18 Girl Wed, Boy Thu

19 Girl Thu, Boy Thu

20 Girl Fri, Boy Thu

21 Girl Sat, Boy Thu

22 Boy Sun, Boy Thu

23 Boy Mon, Boy Thu

24 Boy Tue, Boy Thu

25 Boy Wed, Boy Thu

26* Boy Thu, Boy Thu

27 Boy Fri, Boy Thu

28 Boy Sat, Boy Thu

Note that Boy-Thursday-Boy-Thursday occurs twice (12 and 26). The article (and the quoted comment) incorrectly ignores the second instance because it is not unique, leaving only 27 combinations. The assumption they make is that Boy-Thursday-Boy-Thursday is equally as likely as the other 26 options, which is not true. In fact, the Boy-Thursday-Boy-Thursday is twice as likely because you do not know which son was introduced.

If the second Boy-Thursday-Boy-Thursday is included, the probably that the other child is a boy becomes 14/28, or 50%.