I used to start my

*Product Management*class with a couple of business math problems like the following: let's say we use a given market research technique to measure the value of a product; call the product $i$ and the value $v(i)$. We know -- by choice of the technique -- that the probability that the customer will buy $i$ is given by

$\Pr(i) = \frac{\exp(v(i))}{1 + \exp(v(i))}$.

My question: is this an increasing or a decreasing function of the $v(i)$?

Typically this exercise divided students in three groups:

First, students who were afraid of math, were looking for easy credits, or otherwise unprepared for the work in the class. These math problems made sure students knew what they were getting into.

Second, students who could do the math, either by plug-and-chug (take derivative, check the sign) or by noticing that the formula may be written as

$\Pr(i) = \frac{1}{1 + \exp(-v(i))}$

and working the increasing/decreasing chain rule.

Third, students who had a quasi-intuitive understanding ("grok" in Heinlein's word) that

*probability of purchase*must be an increasing function of

*value*, otherwise these words are being misused.

Ideally we should be training business students to mix the skills of the last two groups: a fluency in basic mathematical thinking and grokking business implications.

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Administrative note: Since I keep writing 4000+ word drafts for "important" posts that never see the light of blog (may see the light of Kindle single), I've decided to start posting these bite-sized thoughts.