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Here's some Python: http://codepad.org/UzSmoxF2

Can it be used for BFS traversal, as in http://codepad.org/BQJLkdCv ?

I don't have any screenshots available offhand, but I have source. It's in C++, and I'm very definitely not a C++ guy, but it works.

No, you're wrong. Look at this. If Mr. Smith has two children, at least one of whom is a boy, it is two times as likely for him to have a boy and a girl than it is for him to have a boy.

Your mistake is in believing that, by virtue of naming one of the boys Peter, the probabilities are magically equalized. They're not. The correct probabilties for your table are:

Peter, Boy = 1/6
Boy, Peter = 1/6
Peter, Girl = 1/3
Girl, Peter = 1/3

You are literally thinking like the girl in this joke.

But ok, I guess the only way to settle this is by experiment. Start sending in women, and I'll get back to you in nine months.

1. "two boys", with probability 1/4
2. "two girls", with probability 1/4
3. "a boy and a girl", with probability 1/2

The last case is twice as likely as either of the others because the two births are independent events (by assumption; in real life there is probably a slight correlation, girl births are more likely than boy births, etc., but it doesn't deviate very far from that).

If you're still unconvinced, look at this: http://codepad.org/kMVsqzyT

Exactly. It seems biased to me, but any arbitrary stopping place would be biased I suppose.

BTW, I posted an earlier comment where I considered all the prime below 1 million and it looks like this:
Begins with 1: 12.21%
Begins with 2: 11.64%
Begins with 3: 11.41%
Begins with 4: 11.14%
Begins with 5: 10.97%
Begins with 6: 10.77%
Begins with 7: 10.74%
Begins with 8: 10.60%
Begins with 9: 10.48%

Here's all primes below 2 million:
Begins with 1: 53.72%
Begins with 2: 6.13%
Begins with 3: 6.1%
Begins with 4: 5.87%
Begins with 5: 5.78%
Begins with 6: 5.67%
Begins with 7: 5.66%
Begins with 8: 5.59%
Begins with 9: 5.52%

And here's all primes below 3 million:
Begins with 1: 36.90%
Begins with 2: 35.52%
Begins with 3: 4.13%
Begins with 4: 4.3%
Begins with 5: 3.97%
Begins with 6: 3.90%
Begins with 7: 3.89%
Begins with 8: 3.84%
Begins with 9: 3.79%

Perhaps an unbiased "test" would be one where you average the results of several tests, picking a random number each time as the stopping place (either the upper limit or the # of primes, probably doesn't matter).

P.S. Here's my source code (uses openssl library): http://codepad.org/0DfZ8uOG