Probability of 2 Different Gendered Children Born in Order

[fog37]

TL;DR Summary

probability in two-child problem

Hello,

This is a classic problem in basic statistics: find the probability of two children of different gender being born, one after the other from the same mother. The sample space is ${GG, BB, GB, BG}$ with B= boy and G=girl.

CASE 1: the event "a boy is born if the first child who was born is also a boy" has the probability $\frac {1}{2}$. The event is $BB$ and the reduced sample space is ${BB, BG}$.

CASE 2: the event "a boy is born and there is at least a boy" has the probability $\frac {1}{3}$ since the reduces sample space is ${BB, , GB, BG}$.

In the first case, the order matters, i.e. $GB \neq BG$ But the order does not matter in case 2 meaning that $GB=BG$

Is my understanding correct?

Thanks!

 

[FactChecker]

CASE 2: the event "a boy is born and there is at least a boy" has the probability $\frac {1}{3}$ since the reduces sample space is ${BB, , GB, BG}$.

This probability is wrong. GB and BG are CONSIDERED the same, in that the order does not matter, but they each have separate probabilities that add to the total. Notice that this case includes all of CASE 1 plus the GB possibility, so its probability must be more that the CASE 1 probability.

CORRECTION: It is not clear to me how these two cases are relevant to the original problem.

 

[fog37]

This probability is wrong. GB and BG are CONSIDERED the same, in that the order does not matter, but they each have separate probabilities that add to the total. Notice that this case includes all of CASE 1 plus the GB possibility, so its probability must be more that the CASE 1 probability.

Well, GB = first a girl then a boy. BG=first a boy then a girl. If the event does not care about the order, wouldn't GB and BG be two different and distinct events?

 

[FactChecker]

On more careful reading, I am having trouble understanding the original question.

This is a classic problem in basic statistics: find the probability of two children of different gender being born, one after the other from the same mother. The sample space is ${GG, BB, GB, BG}$ with B= boy and G=girl.

The answer here is clear. There are 4 equally likely results and 2 of the 4 satisfy the "different gender" condition. So the answer is 2/4 = 1/2.

CASE 1: the event "a boy is born if the first child who was born is also a boy" has the probability $\frac {1}{2}$. The event is $BB$ and the reduced sample space is ${BB, BG}$.

CASE 2: the event "a boy is born and there is at least a boy" has the probability $\frac {1}{3}$ since the reduces sample space is ${BB, , GB, BG}$.

In the first case, the order matters, i.e. $GB \neq BG$ But the order does not matter in case 2 meaning that $GB=BG$

What is the relevance of these cases to the original question?

 

[pbuk]

"a boy is born and there is at least a boy"

These words do not have a clear meaning.

Are you referring to the problem discussed in https://en.wikipedia.org/wiki/Boy_or_Girl_paradox? As you can see from that page, it is very difficult to express problems such as this in a way that is unambiguous.

 

[fog37]

Thank you. I found this website: https://www.theguardian.com/science/2019/nov/18/did-you-solve-it-the-two-child-problem

where my question is formulated more clearly at point 2. The question in 1. considers the order while 2. does not...

1. Mrs Smith has two children. The eldest one is a boy. What’s the chance that both are boys.

If the eldest one is a boy, the youngest one is either a boy or a girl, with a 50/50 chance of each. So the chance both are boys is 1 in 2, or 50 per cent.

2. Mrs Jones has two children. At least one is a boy. What’s the chance that both are boys?

If at least one is a boy, there are three possible equally likely gender-assignations of two siblings. boy-boy, boy-girl, or girl-boy. Only 1 in 3 cases, or 33 per cent are both boys. The lesson here is that when considering equally likely scenarios we must consider birth order. If the birth order of the boy is not specified – i.e. if we don’t know if he is the eldest or the youngest – the probability of two boys drops to 1 in 3.

 

[pbuk]

2. Mrs Jones has two children. At least one is a boy. What’s the chance that both are boys?

If at least one is a boy, there are three possible equally likely gender-assignations of two siblings. boy-boy, boy-girl, or girl-boy. Only 1 in 3 cases, or 33 per cent are both boys.

No, this is not the only possible interpretation of the information given. Consider the question below:

Here is a picture of Mrs Jones and one of her two children.

At least one of her children is a boy: what is the chance that both are boys?

If you confidently answer that as "1 in 3" then consider the question "what is the chance that the child not pictured is a girl?"

 

[pbuk]

So the lesson here is nothing to do with birth order, the lesson is that we can only make statements about probabilities relating to a sample if we clearly define both the population and the method of sampling.

 

[PeroK]

@fog37 you may know this is an infamous problem that is much misunderstood in popular media. And is a potential probability minefield. Your calculations are correct if taken at face value, but (for what it's worth) I agree with @pbuk that a better answer is that ultimately you have to consider how you came by the information.

If you pick a person at random and ask the following questions, with the given answers

How many children do you have? Two.
Are they both girls? No.

(Note that the second question is a clearer equivalent of "is at least one a boy?")

Then it's clear that the probability of the person having two boys is 2/3.

But, if you are simply told that a person has two children and they are not both girls, then you do not know why they gave you this particular information.

There are other twists and subtleties if you start introducing information like names or things the children do.

 

[fog37]

@fog37 you may know this is an infamous problem that is much misunderstood in popular media. And is a potential probability minefield. Your calculations are correct if taken at face value, but (for what it's worth) I agree with @pbuk that a better answer is that ultimately you have to consider how you came by the information.

If you pick a person at random and ask the following questions, with the given answers

How many children do you have? Two.
Are they both girls? No.

(Note that the second question is a clearer equivalent of "is at least one a boy?")

Then it's clear that the probability of the person having two boys is 2/3.

But, if you are simply told that a person has two children and they are not both girls, then you do not know why they gave you this particular information.

There are other twists and subtleties if you start introducing information like names or things the children do.

Wow! I thought it was simpler but I am happy to find out that something apparently simple is instead so twisted.

 

[PeroK]

Wow! I thought it was simpler but I am happy to find out that something apparently simple is instead so twisted.

https://www.physicsforums.com/threads/boy-girl-riddle-conditional-probability.861936/#post-5409436