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.