What is mathematical literacy?

There is a wonderful mathematics joke, usually told about Von Neumann, but sometimes with other mathematicians swapped in, it features the following math problem

A train is traveling west at 70 miles per hour, and a train is traveling east at 80 miles an hour. They started 300 miles apart. A fly decided to challenge itself and, at 100 miles per hour, flew from the west-bound train train to the other, and then back, and then forward, and then back, until the two trains passed each other. (An exercise path that is somewhat akin to what we call “suicides” in high school swim practice.) Question: what is the total distance traveled by the fly?

There are two ways to do this problem.

  1. The brute-force way. The fly and the east-bound train are moving toward each other at 180mph. Starting at 300 miles apart they would meet after 300/180 = 100/60 hours = 100 minutes, during which the fly would have traveled 500/3 miles, and the west bound train 350/3 miles. After the fly turned around, it will be 50 miles (= 500/3 – 350/3) from the west-bound train and they head toward each other at 170 miles per hour, so they would meet after 5/17 hours, during which the fly traveled 500/17 miles and the east-bound train 400/17. Now 100/17 miles apart the fly and the east-bound train head toward each other at 180mph and…

    This eventually leads to an infinite series which one can potentially sum geometrically and arrive at a final answer. Or,

  2. The “smart” way. The two trains move toward each other at 150 miles per hour. It takes 2 hours until they meet. The fly, traveling at 100mph, much have traveled a total of 200 miles during the trip.

The “joke” is usually told with some joker, say dear old Feynman, posing this mathematical problem to some more serious-minded mathematician, often renown for his computational abilities (say Von Neumann). The mathematician would reply instantly 200 miles. The joker would be disappointed and complain that the mathematician had heard this question before and somehow knew “the trick”, upon which the mathematician replies: “What trick? It wasn’t that hard to sum the series.”

I just came back from teaching an exercise course (in Princeton language, a precept), and the official answer to one of the exercises reminded me of the above problem. Here is the question:

You are given a biased coin which lands on tails with probability p. You flip the coin until you get heads, and record the number of tosses. (a) What is the probability that the number of tosses is exactly n? (b) What is the probability that the number of tosses is greater than or equal to n?

Part (a) is standard: for n total tosses, the nth toss is heads and the rest are tails. So the probability is p^{n-1}(1-p), representing n-1 successive tails followed by a single heads.

The official answer to part (b) is this: The probability of number of tosses is at least n is the sum

\displaystyle \sum_{k = n}^\infty P( X = k ) = \sum_{k = n}^{\infty} p^{k-1}(1-p)

which after summing the geometric series we arrive at p^{n-1}.

Here’s what I thought after seeing the question: for at least n total tosses, the first n-1 must be all tails. After which the tosses don’t matter: either one eventually ends up with heads at some finite time, in which case it is counted as some finite k = N event, or one never hits heads, in which cases the probability is vanishingly small. So the probability is exactly the probability of getting exactly n-1 tails in a row, which is p^{n-1}.

Just like the fly problem, one of the solutions brute forces the answer, the other try to “reason” away the computational bits until one is left with a simple computation which can be done quickly in one’s head.

Which of these is mathematical literacy? Read the rest of this entry »