Oz's Question of the day #219 (19/2/10) Maths !

[MaxRyan]

I'm going to have to disagree. By using Coop's exact logic, order of children is irrelevant (i.e. boy-girl is the same as girl-boy). That reduces it to only two choices which is boy-boy or the combination of boy-girl/girl-boy. This is not a permutation. Had the problem stated that the first child was a boy, then Coop's logic would hold true. All it says is one child is a boy.

I agree with you, because it is ambiguous to which child is older both answers are correct according to

Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
Gardner initially gave the answers 1/2 and 1/3, respectively; but later acknowledged[1] that the second question was ambiguous. Its answer could be 1/2, depending on how you found out that one child was a boy. The ambiguity, depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[3] and Nickerson.[4]

 

[Coopdman]

teej21012: This kind of problem can be very confusing. I actually had my answer written as 50% probability and was ready to submit it (stating that Mick had given us the answer in his question, which he's done before, tricky guy!), then my horrible memory of college statistics came to me.

So, take the following as an optional explanation, not an attempt to "prove you wrong" or anything. I'm sure a lot of people, if not all, that missed this one, answered 50%.

First, let's dissect the information:
"My wife and I have have two children.
The probability that the first child is a boy, is 50%.
The probability that the second child is a boy, is also 50%.
I will tell you that one child is a boy.

What is the probability that our other child is also a boy ?"

Facts:
Mick has two children
At least, one child is a boy.
Each child independently has a 50/50 chance of being a boy/girl (true to genetics)

Tricky part:
"The probability that the first child is a boy, is 50%.
The probability that the second child is a boy, is also 50%"

Those two lines are what stumped me at first. This is just showing us that the probability for each child is fair (fair coin flip). Each child on it's own has a 50% chance of being a boy.

The next line is what differentiates the problem from another similar question.

"I will tell you that one child is a boy."
Not, I will tell you the first child is a boy
OR
I will tell you the second child is a boy
This means that the first child could be a boy, OR the second child could be a boy. We don't know which child is the known boy.

So let's look at probability versus possibility. Possibility is what could be true. Probability is, how likely is it for any possibility to be true? OR, how often would this possibility happen?
The possibility of Mick's other child being a boy is 50%

Possibilities:
The known boy could be Mick's first child.
The known boy could be Mick's second child.
The unknown child could be a boy or a girl.
(If you don't like "first" and "second" because of twins etc... you can name them bi-gender names, or simply give them irrelevant names, like Apple and Kiwi)

We can represent this in a chart...

First Child Second Child
Boy Boy
Boy Girl
Girl Boy
Girl Girl <---- we know one child is a boy, so this genetic possibility is irrelevant.

The question:
"What is the probability that our other child is also a boy ?" 2 boys

This distinctly tells us we have to include the known boy into the probability. We can't separate the likelihood of the "other" child being a boy (50%) from the known fact of one child already being a boy. Reworded, what is the probability that Mick has two boys, knowing the facts and possibilities above.

We have three possibilities:
First Child Second Child
Boy Boy
Boy Girl
Girl Boy

(ask yourself, could Mick's "known" child be his first child? his second child?) this is why the order of the children is relevant

Probability:
The possibility of one thing being true / (divided by) ALL possibilities

Three possibilities
Only one possibility includes 2 boys out of the three

The possibility of Mick having two boys / ALL three possibilities

1 /3 = 33.33333333333333% => 33%



Mick: Do you really have two kids?




Looking for honest input:
This is at least the second time I've tried to help explain an answer. I don't want to make anyone upset, so honestly, if you'd rather me keep the thoughts to myself, I'm totally OK with that. If you like the explanation, great and thanks. If you don't, I want to hear from you and I will stop.

 

[teej21012]

@ Coopdman: I see what you're saying but again, order is not important as the problem does not state anything about order. In that sense, boy-girl and girl-boy are one in the same. I had a formula for exact numbers but it's on my work computer and can't get to that til Monday. Maybe Mick will find it in his heart to award both answers since wording of problem is so ambiguous :thumbup:

 

[OZXJR]

Ok guys,we don't want to have any trouble over this question,it is only a game
So I will call the question void.

 

[Kazza]

Geez guys, it's just a bit of fun.

Don't take it too seriously..... :BLAA:

 

[teej21012]

I wasn't getting worked up over it. Just trying to validate my answer. Of course it's all in good fun. I don't think Coop was either. :BLAA:

 

[Coopdman]

No hard feelings here I should probably apologize to fz6sa for voiding his point... sorry buddy.

 

[Senior]

My wife and I have have two children.
The probability that the first child is a boy, is 50%.
The probability that the second child is a boy, is also 50%.
I will tell you that one child is a boy.

What is the probability that our other child is also a boy ?.

Well I PM'd a different answer: 25%
But having just thought about it again, I think I've answered a different question - what is the probabililty of having 2 boy childs (from single births). (i.e 50%x50% = 0.5x0.5 = 0.25 = 25%) or it's 1 combination out of 4
Boy - Boy
Boy - Girl
Girl - Girl
Girl - Boy

So I was wrong, :Im With Stupid: but don't think it's 33.333333333333% either



Anyway, so the question What is the probability that our other child is also a boy ?. is treating the 2 children seperately, not combining in anyway their probabilities, and hence it is either the first or second child, and that probability is already stated as 50%

Just a thought and I'm not going to argue :thumbup:

Roll on tomorrows question . . . .

 

[Fz6Sa]

No hard feelings here I should probably apologize to fz6sa for voiding his point... sorry buddy.

No sweat , we play QoD for the fun of it!!