Probability

• Author: Alison Weir
• Pages: 793-813

793

A. INTRODUCTION

Statistical analysis assumes one has a sample and asks questions about the

populations that could have generated the observed sample. Probability

theory assumes everything is known about a population and asks ques-

tions about potential samples. Probability is littered with paradoxes and

surprising results, including the following examples:

1) The Birthday Problem: How many people should you have at a party if

you want there to be a 50 percent chance of at least two people sharing

the same birthday?

Answer: Twenty-three.

Many people f‌ind this number surprisingly small, but the question

is about pairs of people and twenty-three party-goers allows for 253

dif‌ferent pairings. Enough that there is a 50 percent chance that at

least one pair will share a birthday.

2) The Monty Hall Problem: In the old television game show Let’s Make a

Deal, a contestant is asked to pick one of three doors. Behind one door

is a valuable prize and, behind the other two, a goat. The contestant

gets to pick a door, which remains unopened. The host then opens

one of the other two doors, revealing a goat. The crux of the contest

comes next, when the contestant is asked if he or she wants to keep

the original selection or switch to the remaining unopened door. What

should the contestant do?

CHAPTER 22

Probability

Alison Weir

794 6 Alison Weir

Answer: Switch.

At this point, there are two doors—one has a goat, the other has a

car. Intuitively, one may think that each door has a 50 percent chance

of having the car behind it. So why is switching the right choice? The

key to the answer is that, when the contestant originally chose a door,

there was a one in three chance of winning the prize. The door that

was opened to reveal a goat was not opened randomly. Instead, the

producers of the show made certain to open a door that hid a goat.

As a result, the contestant’s original one-third chance remains one

third. Looking at it another way, the chance of the contestant having

made the wrong choice remains two-thirds, and therefore switching

doubles the chance of winning.

3) The Boy-Girl Problem: Sam has two children. The older child is a girl.

What is the probability that both children are girls? Ali has two chil-

dren. At least one of them is a boy. What is the probability that both

children are boys?

Answer: There is no controversy about the f‌irst question—there

is a 50 percent chance both of Sam’s children are girls. But we cannot

answer the second question because there is a subtle paradox arising

from the two possible ways to approach the problem of Ali’s children.

 Look at all families with two children, one of whom is a boy. The

possible outcomes for these families are, by birth order, boy/boy,

boy/girl, or girl/boy. The probability that both children are boys is

33 percent.

 Look at all boys with exactly one sibling. The possible outcomes are

that the sibling can be either a boy or a girl. The probability that

both children are boys is 50 percent.

B. PROBABILITY

Statisticians and mathematicians express probability as a number between

0 and 1 that represents the degree of belief that a particular event will

occur. An event with probability 1 is certain to occur, and an event with

probability 0 is certain not to occur. Outside of these f‌ields, it is typical

to express the number as a percentage. This is largely a formatting issue.

Where a statistician might express a conf‌idence level of .33, an economist

will be 33 percent conf‌ident.