Dear SIPTAnts,
Here is a small puzzle for you - I give it to my master students - that makes a nice point (it’s not mine, I only created a story around it).
Enjoy, Marco
A casino hosts a slot machine with only two spinning reels, A and B, each one displaying either 0 or 1. You observe the machine for a long time, thus verifying that those reels are independent and uniformly distributed, as they should: P(A)=P(B)=P(B|A)=1/2.
One plays inserting two dollars in the machine, and wins 4 dollars if A=B and nothing otherwise. The game is fair in the sense that on average one wins 2 dollars.
At some point you discover that you can manipulate reel A while it's spinning, thus setting it at the value you like. Are there conditions that enable you to win more than 2 dollars on average by doing so? (Think before reading on.)
Imagine that the machine internally works as follows. It generates two independent uniformly random bits U and V. By a defect of construction, A and B are then set according to the following rules:
- A=1 iff U=V.
- B=1 iff U=1 or (A=1,U=0,V=1). Luckily, this does not affect the fact that the machine is indeed fair: P(A)=P(B)=P(B|A)=1/2.
However, when you manipulate A, its value will no longer depend on U and V; it will just depend on what you decide, say A=1. This will make the clause (A=1,U=0,V=1) true whenever U=0,V=1. Whence B=1 will occur with probability 3/4: 1/2 due to U=1 plus 1/4 due to U=0,V=1.
As a consequence, on average you will win 4*3/4 = 3 dollars, making an average profit of 1 dollar per game.
The moral of the story: causation does not imply correlation - or, prediction does not inform action.