The Two Child Problem (revisited)

[BenVitale]

See article in Wikipedia

And, Column By John Allen Paulos

Scenario 1: A woman has two children, the older of whom is a boy. Given this knowledge, what is the probability that she has two boys?

No problem here, the answer is 1/2

Scenario 2: Assume that you know that a woman has two children, at least one of whom is a boy. You know nothing about this boy except his sex. Given this knowledge, what is the probability that she has two boys?

We have come to accept that the probability the woman has two boys is 1/3, not 1/2.

Here's the kicker:

Suppose that when children are born in a certain large city, the season of their birth, whether spring, summer, fall, or winter, is noted prominently on their birth certificate. The question is: Assume you know that a lifetime resident of the city has two children, at least one of whom is a boy born in summer. Given this knowledge, what is the probability she has two boys?

This appears to be essentially the same as the scenario 2. But instead, they would find that 7/15 of these women have two boys.

An increase in probability from 1/3 to 7/15 !

Doesn't this change in probability seem strange?

I got a bit confused after reading the following problem:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

Answer: the probability of two boys is 13/27

Source: http://alexbellos.com/?p=725

 

[bpet]

... Here's the kicker ...

Nice twist on an old paradox!

Sometimes a diagram helps. The sample space for the day of week version can be visualized on a 14 x 14 grid with the conditional probabilities just being the proportion of red dots (see below).

 

[BenVitale]

bpet,

Thanks. The diagram helps.

So, going back to John Allen Paulos, in the Summer the probability of having boys is higher.
I wonder how did Paulos compute that?

I was looking on the web for seasonal distribution of births of boys, but couldn't find anything. Have you searched for any stats?

All I know is, the ratio of males to females in a population is 107 boys to 100 girls or 105 boys to 100 girls (Fisherian sex ratio), but it's not always the case because of the ratios may be considerably skewed due to abortion, infant mortality, etc.

 

[bpet]

...So, going back to John Allen Paulos, in the Summer the probability of having boys is higher.
I wonder how did Paulos compute that? ...

I thought he was saying the opposite, namely that observing a higher proportion of boys among families with a boy born in summer, does NOT imply that the summer birth ratio is higher. Also, if you want to know where the 7/15 came from, it can be computed by drawing an 8x8 grid similar to the one above.