Mon 25 February 2019

The time I lost faith in Expected Value

Written by Hongjinn Park in Articles

It's a goofy thing to say but I lost some faith/respect in Expected Value after seeing the following problem. Not that EV or anyone cares what I think (but we feel what we feel).

Let's say you're at a Chinese restaurant with ten of your friends. All ten of you order different dishes. Fortunately the table has a Lazy Susan so you can easily share every dish! Let's say you spin the Lazy Susan randomly. Now with probability $1/10$ everyone will get the same dish they started with. Let the RV $X$ be the number of people who get the same dish after a random spin.

Now we have that $E[X] = \frac{1}{10}10 + \frac{9}{10}0 = 1$

I find this to be an interesting result. With every spin, we expect $1$ person to get the same dish. However, in reality either everyone gets the same dish or nobody does. I find this result somewhat misleading. In reality it will be all or nothing, each time.

I suppose you have the same problem with the lottery. If the prize is one trillion and the probability of winning is one in one trillion, then each time you play you have an EV of a dollar. However with odds like that don't expect to ever see money, and if you do you will see a lot of it.

There's also the famous St. Petersburg Paradox which shows how funny EV can be, although infinity can cause weird problems in general.

At the end of the day, perhaps we should not lose faith in EV, but take it with a grain of salt.



Articles

Personal notes I've written over the years.