How to Use Mathematical Expectation in Daily Life (A Demonstration)

Mathematical expectation is one of the single most useful pieces of mathematics ever.

The technical stuff: to calculate it, you multiply the probability of something happening by how much return that thing will give you.  This allows you to compare different paths.  Cool!

I’ll show how this works by example.

I had two possible jobs this week (well, three, but I got called for the third after I’d already committed to one of the others).  One of those jobs was a definite, but it was also a job I would essentially be doing as a favor.  The other job was at my full rate, but I wasn’t sure it was going to happen.  Which to accept?

I’m not going to put the actual amounts down here, because I feel weird talking about what I make online, but the orders of magnitude are right.  To keep the numbers super easy, let’s say it was this:

Favor job: $400

Real job: $4,000

The probability of the favor job happening was pretty much 100%, or close enough.  The real job I wasn’t sure.  But I could estimate based on where the production was in the process and how they were interacting with me.  Let’s say, in my experience, 9 out of 10 jobs that get this far actually happen.  Then I estimate the real job’s probability at 90%.  Thus, my expected return is:

Favor job: $400 x 1.00 = $400

Real job: $4,000 x .90 = $3600

$3600 is way better than $400!  So the clear choice is to take the real job and turn down the favor job.  Even if the real job doesn’t happen, I can be comforted in knowing I’ve made the mathematically correct choice, the choice that was most likely to work out best.  (Yeah, I do find that shit comforting.  So shoot me.)

Okay, let’s say you don’t know the probabilities terribly well.  This can still be helpful.  How low a probability would the real job have to have to give me an expected return as low as the favor job?

Favor job: $400 x 1.00 = $400

Real job: $4,000 x p = $400

It’s pretty easy to see that p would work out to .1, or 10%.  So the real job would have to be so improbable as to only have a 10% chance of happening for both jobs to have the same expected return.  Even if I don’t know the real probability, I might be able to say that I’m pretty sure it’s over 10%.  (And it would have to be below 10% to have a lower expected return than the favor job and finally make the favor job be the better choice.)

So, there’s a quick guide to using mathematical expectation in life.  If you can estimate the rough probabilities of two paths, multiply those by the return, and you’ll get the expected return; compare those and pick the bigger one.  Done!

Man, math makes life decisions so easy.[1]

Not really.↵