To infinity and beyond
September 8, 2011 at 9:59 PM by Dr. Drang
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.
