Probabilities are a curious type of numbers. If properly understood, they can help us see the world around us in an appropriate context. Sometimes these numbers can be counterintuitive and downright confusing and sometimes misleading. This article from The Atlantic presents 5 classic problems in probability that can help us think better in probability and statistics. Some of these problems confounded experts. It pays for anyone to pay attention to these problems.

These 5 problems are the Monty Hall problem, the birthday paradox, Gambler’s ruin, Abraham Wald’s memo and Simpson’s paradox.

One common characteristic to all these problems is that they are in some sense paradoxical. The facts on the surface can lead us down one path to a wrong solution. On the other hand, the correct result can be so counterintuitive that it seems absurd.

Take the Monty Hall problem for example. When it appeared in a column authored by Marilyn vos Savant in Parade Magazine in 1990, it drew a great deal of angry responses from readers, some of whom were holders of PhDs on math and statistics (they said so in their disparaging responses). These experts in math and statistics all claimed that the solution proposed by vos Savant was wrong and she should know better. Some of these angry remarks are repeated here.

These experts were wrong! It turned out that even experts can be confounded by probability numbers too. As a result of the controversy, the Monty Hall problem is a probability problem that is known widely and is covered in most standard introductory texts on probability and statistics. The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brainteaser, a book entirely on the subject of Monty Hall problem, is authored by Jason Rosenhouse (Oxford University Press, 2009).

Refer to the above link for a quick introduction to these 5 problems. Three of the problems have been discussed in several math blogs affiliated with this blog. The following are the links to these blog posts.

- Monty Hall Problem
- Birthday Problem
- Gambler’s Ruin – discussion
- Gambler’s Ruin – deriving the formulas

The following blog posts discuss other classic problems in probability.

- The Dice Problem
- The Problem of Points
- The Coupon Collector Problem
- Two more discussions of the Coupon Collector Problems are this one and this one.
- Using the digits of Pi as random digits
- Two more discussions on the Birthday Problem: here and here.