Talking through Christopher Long’s neat probability problem with kids part 2

Yesterday we talked a bit about this really neat probability problem that Christopher Long posted on Twitter:

Choose positive integers a,b,c,d "at random" (1..N, uniform mass function, take limit). What's the probability that gcd(a,b) = gcd(c,d)?

— Christopher D. Long (@octonion) September 10, 2015

Here’s a link to that conversation:

Talking through Christopher Long’s probability problem with kids (part 1)

We finished up the conversation today.

Yesterday we ended after talking about why the probability that two randomly chosen integers will share no common divisors is . Today we revisited that conversation and also discussed why picking two random positive integers is actually a little bit of a hard thing to do:

 

Next I wanted to move on to discuss the last part of the solution to the original probability problem, but a question about fractions came up and we had a short conversation about which fraction was larger 36 / 81 (which arose in the problem from the approximation that , or the fraction . So, this strange probability problem gave us a surprising way to talk about fractions 🙂

 

After the short conversation about fractions we returned to the problem. We now know that the probability that the two randomly selected pairs of integers will both be relatively prime is But what about the probability that both pairs will have a common divisor of 2?

The answer to this question is a little subtle, but it is the key to solving the original problem. Thinking about this question led to a great conversation about primes and what GCD means.

 

Now that we had a little bit better of an understanding about primes and GCD, we dove into how to think about the problem where the two pairs of integers share a common divisor of 2. In this video we talk about why the probability that two randomly selected integers will have a the greatest common divisor of 2 is 1/4 of the probability that they are relatively prime.

 

Finally, we arrive at the solution to the original problem: the probability is 2/5. Surprise!! I bunch of powers of all cancel at the end. Super fun problem.

 

The original problem is obviously way too difficult for kids, but walking them through the solution was really fun. Along the way we got to talk about fun concepts like infinity, fractions, primes, divisibility, and even some really advanced topics like how does show up in this problem?

I definitely wouldn’t do a project like this one too often, but once in a while an advanced problem actually has some pretty neat stuff for kids hiding inside of it.