Try Everything: the Monty Hall Problem and the Bayes’ Theorem
You have a job but you don’t quite like it. Will you stay in your current position forever or will you explore? It is a hard question, for the cost to switch jobs, and the uncertainty of new positions. Life is short and math has told us how to decide!
There is a classic Monty Hall problem. Suppose there are three closed doors in front of you and there is a prize behind one of the doors. You get the prize if you select the correct door. There is a host, who knows behind which door the prize is. You pick one door first. Before you open the door you selected, the host will open another door without the prize behind it. Now should you stick to your choice or switch?
At the first glance, it seems the host does not help at all, and the two strategies make no difference. But they are indeed very different!
Before the host rules out one door, the probability you select the correct door is 1/3. If you stick to your selection, this probability does not change. But if you switch, the probability you win the prize is 2/3. Because with probability 2/3, the prize is behind one of the doors you did not select, and the host ruled out one of them.
By switching your choice, your winning probability doubles! This is surprising, even to some experts who never thought about this problem before. This is how the host is helpful, or the power of data.
Yes, it is harder to switch jobs than switching your selection of doors. But it is worth it, if you can rule out some bad choices, acting as the host yourself.
The Monty Hall problem is a very classic problem in statistics classes, when professors teach Bayes’ theorem. From the Bayesian point of view, there is a prior probability of 1/3 for winning the prize. After you observe the door with nothing behind, you can calculate the posterior probabilities of winning for sticking to your choice and switching. And the result is exactly the same as I explained.
In the machine learning area, it is also well-known that bad examples help us train good models, as well as the good examples. All information helps, if you know how to leverage.
