On various social media sites I’ve seen the following hypothetical scenario bandied about:
You are given the option to get $50,000 automatically or flip a coin for a 50% chance to win $50 million. Which option would you take?
Most people, it turns out, will opt for the $50k, which caused a lot of internet randos to say that people clearly aren’t very numerate, because the best return on investment is Option 2, which will on average give you $25 million.
A lot of you can probably already see the flaw with this basic argument about which choice is best, but it reminded me of a classic “paradox” in probability theory known as the St. Petersburg paradox. This felt like a good excuse to talk about that paradox and connect it back in a roundabout way to the scenario described above!
The St. Petersburg paradox doesn’t directly have to do with St. Petersburg; it just so happens that it was invented by Nicolas Bernoulli and analyzed by his brother Daniel Bernoulli in Commentaries of the Imperial Academy of Science of Saint Petersburg.
The paradox may be formulated as follows: You are invited to play a game of chance. You will flip a coin until it turns up heads, and every time you flip tails your winnings double. That is, if you flip heads immediately, you get $2, if you flip heads on the second toss you get $4, on the third toss you get $8, and so forth. In principle, if you keep flipping tails, you could win an astounding amount of money. The question is: what is the most you should be willing to bet to play such a game?
For those with some math knowledge, the natural thing would be to calculate the average winnings. The chance of flipping heads on the first toss is 1/2, the chance of flipping it on the second toss is 1/4, the chance of flipping it on the third toss is 1/8, and so on. If we label those chances 
The average winnings are infinite! It would superficially seem like it is a game where betting any amount of money will produce a great return. So the game master offers to let you play for $50 a shot, which you accept, but after a number of tries you find that you’ve lost all the money in your pocket. So how is it that you can lose a game where the average winnings are infinite?
There are a number of different ways to look at resolving the paradox. The simplest way to look at it is that the concept of an “average” itself is an oversimplification of a problem that throws out a lot of information. For instance, what is the chance that you actually win money on any particular play? If we use $50 as our buy-in, and note that the progression of winnings is 2, 4, 8, 16, 32, 64, etc. we would have to roll at least five tails in a row to break even. The chance of doing that is
To put it another way, you’d have to play at least 32 times to have a reasonable chance to win anything, and that win will likely not get you back the money you’ve lost. The “infinite winnings” of the game comes from the unstated assumption in the averaging process that an infinite number of attempts will be made. To put it another way, there is an infinitely small chance that you could win an infinite amount of money, and you’ll only succeed if you try an infinite number of times!
This to me is the best resolution of the paradox: any finite number of plays of the game will almost certainly be losers, and the naive averaging process does not account for that. All of us have a finite amount of money to bet with, so when we play the game we will run out of money before we hit on that lucky, vanishingly small, winning streak.
This brings us back to the scenario mentioned at the beginning of the post and why I thought of the St. Petersburg paradox in the first place! The scenario is also given devoid of real world concerns. Of course, if I didn’t have to worry about money at all, then taking the bet for $50 million would be perfectly fine. But I do have to worry about money, and $50k would be a guaranteed life-changing income, which is much more appealing than the chance to walk away with nothing at all. Just like in the St. Petersburg paradox, the internet scenario analysis using an average doesn’t take into account real-world considerations.
I like to use the St. Petersburg paradox to highlight how averages can be misleading. Sticking to money examples, it’s like the classic case of asking what the average wealth of ten people in a room is when one of them is a billionaire. The average wealth is $100 million, even though 9 of those people have almost no money at all. The St. Petersburg paradox shows that averages can not only be misleading, but can in a sense be infinitely misleading!
Skulls in the Stars 