Chapter 1: What Is Probability?

1.1 "Probability" in everyday speech

We use the word "probability" every day:

• "They say there's a 60% chance of rain tomorrow."
• "What are the odds of winning this lottery?"
• "What's the probability of rolling a 6 on a die?"

All of these try to express how likely something is to happen, before it happens or before we know the result.

Probability is, in short, a way of putting a number on uncertainty. Mathematically, it is treated as a rule that assigns a number between 0 and 1 to each event (a collection of outcomes); Chapter 3 introduces this through sample spaces and the axioms of probability.

1.2 The intuitive meaning of probability

Imagine rolling a die once.

The result is one of 1, 2, 3, 4, 5, 6. If we assume each face is equally likely to come up, then:

• "the result is 1" is one outcome out of six possibilities
• so the probability that 1 comes up is $\frac{1}{6}$

This is the basic way to compute probability when there are finitely many equally likely outcomes.

1.3 Why probabilities lie between 0 and 1

Why is a probability always between 0 and 1? Consider the case of finitely many equally likely outcomes, where

$$\text{probability} = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$

• The number of favourable outcomes is at least 0 (it cannot be negative).
• The number of favourable outcomes is at most the total (it cannot exceed it).

Therefore:

• The minimum value is $\frac{0}{\text{total}} = 0$ (the event never occurs).
• The maximum value is $\frac{\text{total}}{\text{total}} = 1$ (the event always occurs).

When outcomes are not equally likely (such as the loaded die in Section 1.5), probabilities are still treated as numbers between 0 and 1. Under the frequentist interpretation (Section 1.6), a probability is a proportion of occurrences, which also lies between 0 and 1.

Note that “probability 0 = never happens” and “probability 1 = always happens” hold when there are finitely many outcomes, as in this chapter. With infinitely many outcomes—for example, choosing a real number between 0 and 1 without bias—the probability of picking exactly 0.5 is 0, yet it is not impossible. Chapter 3 discusses this further.

1.4 Notation for probability

In mathematics, probability is written as:

Examples

• $P(\text{rolling a 1 on a die}) = \frac{1}{6}$
• $P(\text{getting heads on a coin}) = \frac{1}{2}$

1.5 What does "equally likely" mean?

The dice example in Section 1.2 made an important assumption:

Every face is equally likely to come up.

Why is this reasonable?

• An idealized die is a cube; every face has the same shape and area.
• Its centre of mass is at the centre, with no bias.
• So it is natural to assume that no face is favoured over another (a real die may have slight biases from its manufacture or the way it is thrown).

When physical or structural symmetry lets us assume that every outcome is equally likely, we say in mathematics that the outcomes are "equally likely."

When this assumption fails, simple counting cannot give the probability. For instance, a loaded die (with a biased centre of mass) makes some faces more likely than others.

This introductory chapter focuses on cases where outcomes are equally likely.

1.6 Probability and proportion

Probability is closely related to proportion.

If you roll a die 1000 times, roughly how often will 1 come up?

This is the expected count; the actual count in an experiment fluctuates around it. For 1000 rolls, it falls between 144 and 190 with about 95% probability.

In the frequentist interpretation, probability is understood as the long-run proportion.

This is one of the central interpretations of probability (the frequentist interpretation). There are other interpretations — notably the Bayesian view of probability as a degree of belief — but this introduction focuses on the frequentist view.

1.7 Chapter Summary

FAQ

Exercises