AP Statistics Lectures
by Arnold Kling

Introduction

Until relatively recent times, people did not think of probability as a branch of mathematics. Mathematics was supposed to be about certainty. Therefore, it did not seem to apply to phenomena such as tossing coins, rolling dice, or life expectancy.

Philosophically, people always had the option of refusing to believe in chance. Instead, they could divide phenomena into those that they can predict and those that they cannot predict. For a long time, people thought that it was futile to try to calculate the unpredictable. Even today, most people prefer to use rough intuition to deal with chance, rather than go through the effort to calculate exactly.

According to Peter L. Bernstein, in his book Against the Gods: The Remarkable Story of Risk, humans played games of chance for thousands of years without ever working out exact probabilities. For example, one popular game was to have a player roll a six-sided die ("die" is singular for "dice") four times. If any roll is a six, the roller wins. Otherwise, the opponent wins. People thought that this was a reasonably fair game (the roller and the opponent seemed to win about equally often), but no one worked out the exact odds.

The classic problem that Pascal and Fermat solved was somewhat like the problem that would occur if a major sports championship, such as the World Series or the Stanley Cup, could not be completed.

Imagine that one team was leading three games to two when play had to be stopped. What is the fairest way to divide the prize money, given that ordinarily the team that wins four games gets the entire prize?

Pascal and Fermat took the view that the prize should be divided between the two teams in proportion to their probability of winning. Assume that for each subsequent game each team has a fifty-fifty chance of winning that game. The team that has already won three games will win the championship a higher proportion of the time. Pascal proceeded to show how to calculate that proportion precisely.

Around 1650, Blaise Pascal and Pierre de Fermat solved a problem related to gambling that had been posed 150 years before. From this point forward, the concepts of probability and statistics were given a mathematical treatment.

You are So Random

Our textbook begins its discussion of probability with a section on "randomness." The word "random" is an adjective. What does it modify?

When people ask me why I am teaching, I say that it is because my daughters kept telling me, "Dad, you are so random." I thought they meant that I should give a class in statistics.

However, a mathematician would not say that you can use the word "random" to describe a person. Nor can you use "random" to describe a thing. The term "random" can only be used to describe a process.

A process is random if it can be executed repeatedly with different outcomes. Examples would be rolling a die, spinning a dreidel, or shooting an arrow at a target.

The opposite of random is deterministic. If a process is deterministic, then the outcome of the process is always the same. Every morning, the sun rises in the East. Every time you drop a rock, it falls to the ground.

Whether or not a process is deterministic may depend on how you define the outcome of the process. For example, if you define the outcome of dropping a rock as "landing on the ground or staying in the air," then it is deterministic--it always lands on the ground. On the other hand if you define the outcome as "the number of centimeters away from my foot that the rock lands," it is random.

A process can be random without all outcomes being equally likely. The Red Sox usually win when Pedro Martinez pitches. However, they do not always win. Therefore, the outcome is random.

QUESTION: After an election, there is sometimes a recount. Should the recount always have the same outcome? Is the recount process random or deterministic?