Probability problems

[Mark R]

Problem 1
Alice has 2 children. One is a boy. What is the probability that the other child is also a boy?

-------
Problem 2
Bob has 2 children. One is a boy, born on a Friday. What is the probability that the other child is also a boy?

Enjoy.

(This problem was in this weeks New Scientist magazine - so if you really want to know the answers, you can read the article. But really, these problems are so ingenious that it's worth trying to work them out first, and not read the spoilers).

 

[chuckywang]

1/2

 

[DrPizza]

A) 2/3? (edit: ugh, I just realized I was thinking right, but did girl instead of boy.)

B) Not sure how Friday changes anything. Thought about it, figured it out & came back in the house to check the answer. Got it right.

 

[Mo0o]

1: 1/3
2: hmm dont know this one, i can't figure out why friday would add to it

 

[EarthwormJim]

1: 1/2. The second child's birth is an independent event from the first. The question would have to be reworded for it to be 1/3.

2: 1/2.

 

[MovingTarget]

http://en.wikipedia.org/wiki/Conditional_probability

 

[Leros]

1/2
1/2

They're both independent as somebody else said. Does statistics agree?

 

[KoolAidKid]

http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

 

[DrPizza]

Interesting twist #3 to this question. From a large number of families each having two children, Alice's family was chosen. One of her children was selected at random. It's a boy. What's the probability that the other child is a boy?

Now, it's 1/2.
----

Same problem, restated: Get 4 cards. One card is black on both sides, one card is green on both sides, and two cards are black on one side and green on the other. (black/green = boy/girl) Put them into a hat & mix thoroughly. Pull out one of the cards and look at ONLY one side. Every one of the 8 sides has an equal probability of being looked at. If you happen to be looking at a black side, it's one of the 4 black sides. 2 of those black sides have black on the opposite side, and 2 of those black sides have green on the opposite side.

 

[Syringer]

Trick question, Alice and Bob actually live in China and are limited to one child, hence there is no second child.

 

[Leros]

Trick question, Alice and Bob actually live in China and are limited to one child, hence there is no second child.

If Eve kidnaps their first child, are they allowed to have a second?

 

[Hacp]

When you say one is a boy do you mean child one is a boy? You're confusing all of us, there is not enough information to answer your question.

 

[Ruptga]

I don't see what the gender of the first has to do with the gender of the second. And that Friday thing is just a red herring.

 

[Schfifty Five]

When you say one is a boy do you mean child one is a boy? You're confusing all of us, there is not enough information to answer your question.

I believe he is saying "...of the two children, at least one of them is a boy." If he was talking about the first child being a boy, he probably would have said "first".

Having said that, if one child is a boy, the probability of the other child being a boy is 1/3.

The possible 2 children combinations are:

Boy, Boy
Boy, Girl
Girl, Boy
Girl, Girl

We know that the problem states that one of the children is a boy, therefore, GG is eliminated from the possible outcomes. That leaves BB, BG, GB, and since we are looking for BB, it's going to be 1/3.

 

[Schfifty Five]

I don't see what the gender of the first has to do with the gender of the second. And that Friday thing is just a red herring.

I'm not sure about the Friday thing either yet, but knowing that one of the two children is a boy gives us some information about the combinations of children (see my post above).

 

[her209]

I believe he is saying "...of the two children, at least one of them is a boy." If he was talking about the first child being a boy, he probably would have said "first".

Having said that, if one child is a boy, the probability of the other child being a boy is 1/3.

The possible 2 children combinations are:

Boy, Boy
Boy, Girl
Girl, Boy
Girl, Girl

We know that the problem states that one of the children is a boy, therefore, GG is eliminated from the possible outcomes. That leaves BB, BG, GB, and since we are looking for BB, it's going to be 1/3.

This.

And as to why (Boy, Girl) and (Girl, Boy) are not treated the same, its because of the order of the birth matters.

 

[EarthwormJim]

This.

And as to why (Boy, Girl) and (Girl, Boy) are not treated the same, its because of the order of the birth matters.

Order does not matter because it is not asked in the question. The question is asking for an attribute, not a sequence.

BG and GB have the same number of boys, one.

 

[her209]

Order does not matter because it is not asked in the question. The question is asking for an attribute, not a sequence.

BG and GB have the same number of boys, one.

No, it matters in this case. The probability of a woman having a one boy and one girl is twice as high as having boy boy or girl girl. Going back to Question #1, the chances of a woman having a girl when she already has a boy is 2x of having another a boy, i.e., 2/3.

 

[EarthwormJim]

No, it matters in this case. The probability of a woman having a one boy and one girl is twice as high as having boy boy or girl girl. Going back to Question #1, the chances of a woman having a girl when she already has a boy is 2x of having a boy, i.e., 2/3.

One birth is fixed, the only variable in this case is the unknown child which can be either a boy or a girl.

You cannot include GG since it is impossible in this case.

Possible events are CG CB or BC GC where C is the known boy.

The question would have to be reworded more like "What is the probability that a women will have two boys."

 

[her209]

One birth is fixed, the only variable in this case is the unknown child which can be either a boy or a girl.

And that's what messes up people.

The question would have to be reworded more like "What is the probability that a women will have two boys."

1/4

 

[EarthwormJim]

And that's what messes up people.


1/4

Please list the possible events then...

 

[her209]

Please list the possible events then...

A woman can have in the following order:

Boy then boy
Boy then girl
Girl then boy
Girl then girl

Since we know that one of the children is a boy, we are left with 3 possibilities out of that list, i.e., the possibility of a woman having a combination of (girl,boy) is twice as likely than it is for her to have (boy, boy).

 

[Hacp]

It isn't 1/2. You forget that there are three choices. Boy, girl, and transgender.

 

[EarthwormJim]

A woman can have in the following order:

Boy then boy
Boy then girl
Girl then boy
Girl then girl

Since we know that one of the children is a boy, we are left with 3 possibilities out of that list, i.e., the possibility of a woman having a combination of (girl,boy) is twice as likely than it is for her to have (boy, boy).

But the question doesn't ask what a woman can have, it asks what are the chances that the unknown child is a boy.

Two boys can be two possibilities: the known was born first, the unknown second; or the unknown first, the known second.

B - b
b - B
B - g
g - B

 

[mjrpes3]

Problem 1
Alice has 2 children. One is a boy. What is the probability that the other child is also a boy?

-------
Problem 2
Bob has 2 children. One, and only one, is a boy born on a Friday. What is the probability that the other child is also a boy?

Crappy ambiguity issue fixed. Jeesh, was that so hard?