Mr Honner

Years ago, one sunny Sunday afternoon, I was driving home from visiting friends at college and received a speeding ticket.  I didn’t realize it at the time, but calculus played an important role in my citation.

You see, this was no ordinary speeding ticket, the kind where a police officer paces the offender or uses radar to measure a vehicle’s speed.  My speed was calculated from an airplane high above the road.  And the Mean Value Theorem clinched the case.

Aerial speed enforcement works like this:  large marks painted on the road divide the highway into quarter-mile intervals.  A pilot flying overhead uses a stopwatch to time a suspected speeder from one mark to the next.  Say the pilot records a time of 12 seconds; a simple calculation converts one quarter mile per 12 seconds into 75 miles per hour; this information, the average speed on this interval, is radioed to the police on the ground who then stop and ticket the driver.

What I didn’t realize at the time was how crucial calculus is in all of this.

A fundamental theorem of calculus, the Mean Value Theorem (MVT), relates the average rate of change of a function with the instantaneous rate of change of the function.  Suppose we have some function of time, , and suppose that we know the value of this function at two times, say and .  The average rate of change of  between and is

The MVT tells us that, as long as  is a differentiable function, then at some time between  and , say at t = c, the instantaneous rate of change of  must have been equal to the average rate of change of  from  and .  That is,

where is the derivative of , the instantaneous rate of change of .

What does this have to do with my speeding ticket?  Well, as I’m moving along the highway in my car, the pilot records two values of my position function, , at two different times,  and .  The pilot then computes my average speed

Here’s where calculus comes in.  The Mean Value Theorem says that, at some point between those two times my instantaneous speed must have been equal to my average speed.  If my average speed was above the legal limit, then at some time between  and , my instantaneous speed must have been above the limit, and at that moment, I was guilty of speeding.

I wonder if it would have helped to argue that my position function wasn’t differentiable!

When I visited Mongolia 15 years ago, a young man approached us on the streets of Ulan Bator and offered to show us around. His name was Soyoloo, and we spent a few fun days hanging out with him.

I have many fond memories of Mongolia, so this NYT piece on Ulan Bator’s underground homeless population immediately caught my attention. The first picture is of a man named Soyoloo.  I wondered if this was the same Soyoloo I met so long ago.

How likely is that?  A naive estimate looks like this:  half of the one million people in Ulan Bator are men, so there’s a 1 in 500,000 chance that a randomly-selected (that is, randomly photographed) man from Ulan Bator is the man I met 15 years ago.  At 0.0002%, this does not seem very likely.

But here’s where conditional probability comes in.  The man in the photograph isn’t just any randomly-selected man; he’s a randomly-selected man named Soyoloo.  What I really want to know is, given the condition that a randomly selected man from Ulan Bator is named Soyoloo, what is the probability that he’s the man I met 15 years ago?

Naturally the answer depends on how many men in Ulan Bator are named Soyoloo.  ”Soyoloo” doesn’t appear on any list of most popular Mongolian names, so I am going to estimate that at most 0.5% of Mongolian men are named Soyoloo (this would be roughly equivalent to ‘Andrew’ in the US, the 35th most popular male name).  This means that at most 2,500 men (0.5% of 500,000) in Ulan Bator are named Soyoloo.  Therefore, if a man in Ulan Bator named Soyoloo is selected at random, there is a 1 in 2,500 chance (0.04%) it’s the man I met.

This is the power of conditional probability.  The man in the photograph and the man I met share a common characteristic:  being named Soyoloo.  Knowing this condition greatly alters the probability that they are the same person.

In fact there’s another characteristic the two men share:  they are both comfortable striking up relationships with foreign strangers, whether it’s me or a photojournalist.  Let’s call this characteristic gregariousness.  Now the question is, what is the probability that a randomly-selected gregarious male resident of Ulan Bator named Soyoloo is the person I met 15 years ago?

Well, how many gregarious Mongolians are there?  Let’s say 5% of all Mongolians are gregarious.  Then there are roughly 125 gregarious Mongolian men named Soyoloo in Ulan Bator, and thus, a 1 in 125 (0.8%) chance that the man in the photo is the same man I met.  Not a lock, but not a longshot, either!

This probability may seem surprisingly high, given the distance in both time and space,  but I have a feeling it’s him.  His eyes look warmly familiar.

One thing I’d like to see more of in math education is an appreciation for practical consumer math.  For example, suppose your favorite cereal brand unceremoniously starts arriving in a smaller box.

Students should learn to be on the lookout for things like this.  They should develop a quantitative curiosity, exploring the various ways that quantity can be disguised.

And ultimately be able to put together a real quantitative analysis that helps them make good decisions.

This is math that everyone can use!