### Conditional Probability, the answers

Apr. 11th, 2008 05:52 pm**lederhosen**

Answers to the other day's questions below the cut.

(a) 1/4, (b) 2/4, (c) 1/4. An easy one to warm up.

If you interpret the information imparted here as 'at least one child is a girl' (which is what I intended, and how everybody who answered seems to have interpreted it), the correct answer is (a) 2/3, (b) 1/3. However, I realised afterwards that this one is ambiguous (with some impact on #3 and #4); more on this below.

The answer is still 1/3. You already know that the owner of the SC books is a girl; knowing that she has a girl's name isn't providing any extra information about her.

Still 1/3. This is where things really get really tricky.

Supposing I had no prior knowledge beyond "two kids", and asked "what sex is your oldest child?" and been told "girl" - or even "girl called Mary" - the odds of the younger child also being a girl are 50/50, because we don't have

But in this scenario we

1. Oldest child is a girl (and owns the SC book), youngest is a boy (1/3).

2. Oldest child is a girl, youngest is also a girl (1/3).

3. Oldest child is a boy, youngest is a girl (and owns the SC book) (1/3).

Case 2 breaks down into two sub-cases:

2a. Both children are girls, oldest one owns the Saddle Club book (1/2*1/3=1/6).

2b. Both children are girls, youngest one owns the Saddle Club book (1/6).

Suppose you ask "Is your oldest child a girl?" and the answer is "yes". You've just eliminated case 3, leaving you with two equally-likely cases, in one of which the younger child is a girl and in the other it's a boy.

But if you're told "the oldest one is a girl and owns the Saddle Club book", you've eliminated case 3

As discussed above, 1/2 for both.

1/2.

As discussed above, BG families are twice as common as GG families. But because a GG family has two daughters - who are unlikely to have the same name - it is twice as likely to have a 'Mary', and hence twice as likely to show up on that list as a family with only one girl, balancing out the scarcity of such families.

Now, getting back to question 2... the problem with the way I worded it is that you gained that information by a chance process that gave you a piece of evidence. We know that BG families are twice as common as GG families - but is that chance process equally likely for both families? It might be quite reasonable to suppose that GG families will have

The key here is that figuring out the probabilities doesn't just depend on the knowledge that a certain child is a girl; the fact that you

*1. A friend mentions that they have two kids. With no other information, what are the odds that they have (a) two boys, (b) one boy and one girl, or (c) two girls? (For the sake of these problems, assume that half of children are boys, half are girls, and people don't have a natural tendency to children of one sex or the other.)*(a) 1/4, (b) 2/4, (c) 1/4. An easy one to warm up.

*2. Looking at one of their many bookshelves, you spot a Saddle Club book, which (applying gender stereotypes) you may take as indication that at least one of their children is a girl. Based on this information, what are the odds that they have (a) one boy and one girl, or (b) two girls?*If you interpret the information imparted here as 'at least one child is a girl' (which is what I intended, and how everybody who answered seems to have interpreted it), the correct answer is (a) 2/3, (b) 1/3. However, I realised afterwards that this one is ambiguous (with some impact on #3 and #4); more on this below.

*3. You mention the Saddle Club book to your friend. He replies "Yeah, that's Mary's." What are the odds that both their kids are girls?*The answer is still 1/3. You already know that the owner of the SC books is a girl; knowing that she has a girl's name isn't providing any extra information about her.

*4. As above, but the reply is "Yeah, that's Mary's, she's my eldest."*Still 1/3. This is where things really get really tricky.

Supposing I had no prior knowledge beyond "two kids", and asked "what sex is your oldest child?" and been told "girl" - or even "girl called Mary" - the odds of the younger child also being a girl are 50/50, because we don't have

*any*information about that child. In both that scenario and this one, we end up in possession of the same fact: the oldest child is a girl (called Mary, but at present that's irrelevant).But in this scenario we

*also*know that this particular girl is the one who owns the SC book, and that changes the odds. Before you ask about the book - knowing only that at least one of the children is a girl - the probabilities are:1. Oldest child is a girl (and owns the SC book), youngest is a boy (1/3).

2. Oldest child is a girl, youngest is also a girl (1/3).

3. Oldest child is a boy, youngest is a girl (and owns the SC book) (1/3).

Case 2 breaks down into two sub-cases:

2a. Both children are girls, oldest one owns the Saddle Club book (1/2*1/3=1/6).

2b. Both children are girls, youngest one owns the Saddle Club book (1/6).

Suppose you ask "Is your oldest child a girl?" and the answer is "yes". You've just eliminated case 3, leaving you with two equally-likely cases, in one of which the younger child is a girl and in the other it's a boy.

But if you're told "the oldest one is a girl and owns the Saddle Club book", you've eliminated case 3

*and*case 2b. The remaining cases are case 1 and 2a; 1 is twice as likely as 2a, so the odds of the second child being a girl are only 1/3.*5. Different friend, same dilemma - two kids, you don't recall their sexes. Being sneaky, you ask "Would your oldest like to come to my kid's birthday party?" and the response is "Yes, she'd love to." What are the odds that both their kids are girls?*

6. As above, but the response is "Mary? Yes, she'd love to."6. As above, but the response is "Mary? Yes, she'd love to."

As discussed above, 1/2 for both.

*7. You find a Saddle Club book lying on the ground, inscribed 'To Mary'. You ask around the local schools and in a flagrant breach of privacy they give you the addresses of several dozen Saddle-Club-age 'Mary's in the neighbourhood. You go to the first house on the list and, by examining shoes again, deduce that the family has two children. What are the odds that both are girls?*1/2.

As discussed above, BG families are twice as common as GG families. But because a GG family has two daughters - who are unlikely to have the same name - it is twice as likely to have a 'Mary', and hence twice as likely to show up on that list as a family with only one girl, balancing out the scarcity of such families.

Now, getting back to question 2... the problem with the way I worded it is that you gained that information by a chance process that gave you a piece of evidence. We know that BG families are twice as common as GG families - but is that chance process equally likely for both families? It might be quite reasonable to suppose that GG families will have

*more*'girl books' than BG families, making it more likely that you'll spot one while looking at their shelves - which pushes the odds back towards 50/50, much as in #7.The key here is that figuring out the probabilities doesn't just depend on the knowledge that a certain child is a girl; the fact that you

*got*that knowledge may be information in itself.