# 3

# Tiny Probabilities: Why Are They So Hard to Escape?

## PROBABLE IMPROBABILITIES

Chapter 2 dealt with probability problems where the answers may seem surprising and counterintuitive, at least at first. In this chapter, we will continue on this path and focus on a particular oddity: Why do unlikely things happen all the time? Why do people keep winning the lottery and getting struck by lightning when the chances of such events are so minuscule? We have already touched on this in the birthday problem. On page 63 we saw that, in a group of 100 people, we can be almost certain that at least two people have the same birthday (the probability is 0.9999997). If we consider two particular individuals, the probability that they share a birthday is 1/365 (the first can have any birthday; the second must match), which is not very much, only about 0.3%. However, with 100 people, we have pairs to try for a match, and it is very likely that we finally find one.

The birthday problem illustrates the general idea that although something is highly improbable in the *particular* case, it can still be highly probable in *general* because there are so many particular cases to try out. For another example, consider the state lottery game “Pick 3” where you are asked to match three numbers drawn from 0 to 9, in the order they are drawn or in other words to match one of the thousand combinations ranging from ...