To infinity and beyond

Here’s a math problem my sixth-grader came home with last night.

You have a bottomless pit with \(x\) marbles in it. Two of the marbles are black. You reach in and pull out two marbles at random. If \(p(x)\) is the probability that both marbles are black, what is

\[\sum_{x=3}^\infty \;p(x)\]

I sure don’t remember doing infinite series when I was in sixth grade. I think the teacher was looking more for the students’ reasoning than a numerical answer, but after my son went to bed, I couldn’t stop myself from writing a little program to get the answer.

The number of ways you can draw two marbles from a set of \(x\) marbles is the binomial coefficient:

\[{x \choose 2} = \frac{x!}{2! (x - 2)!}\]

Since there’s only one way to get two black marbles in a draw,

\[p(x) = \frac{1}{{x \choose 2 }} = \frac{2! (x - 2)!}{x!}\]

We could write a very simple program that sums up a long series of such terms, but it wouldn’t be very efficient to keep calculating factorials again and again. A better way is to use this recurrence relation,

\[{x \choose 2} = {{x-1} \choose 2} + (x-1)\]

and to recognize that our starting point is

\[{2 \choose 2} = 1\]

Here’s a quick Python program to estimate the answer:

python:
 1:  #!/usr/bin/python
 2:  from __future__ import division
 3:  
 4:  total = 0
 5:  lastdenominator = 1
 6:  for x in range(3, 1001):
 7:    denominator = lastdenominator + (x - 1)
 8:    p = 1/denominator
 9:    total += p
10:    lastdenominator = denominator
11:  
12:  print total

This gives a total of 0.998. Increasing the upper bound from 1,000 increases the total but never puts it over 1. An upper bound of 1,000,000 gives a total of 0.999998. Pretty clearly, we’re converging to 1.

This is, of course, nothing like a proof. But simple numerical experimentation like this is very compelling and, if you’re of a certain mindset, can seem more real and significant than a formal mathematical derivation.

I don’t know how my son’s teacher covered this problem in class, but I have noticed that my kids do a lot more estimating than I ever did. Roots of equations, for example, are often found by plotting them on a graphing calculator and zooming in to get the coordinate where the function crosses the abscissa—a technique that was impossible when I was in school. There is, no doubt, something lost when kids spend less time on exact methods, but I think the gains made in getting them to think mathematically and make good estimates is worth it.