What does the notation P(A) mean?

P stands for Probability and A denotes the event of interest. P(A) is read as 'the probability that event A occurs.' Examples: P(rolling a 1 on a die) = 1/6, P(getting heads on a coin) = 1/2.

What is the frequentist interpretation of probability?

It interprets probability as the long-run relative frequency of an event under repeated trials. For example, rolling a die 1000 times will yield approximately 1000 × 1/6 ≈ 167 ones. There are other interpretations (notably the Bayesian view as a degree of belief), but this introduction focuses on the frequentist interpretation.

Chapter 1: What Is Probability?

Q: Why is a probability always between 0 and 1?

Probability is computed as 'number of favourable outcomes divided by total number of outcomes.' The numerator is non-negative and at most equal to the denominator, so the value is always between 0 and 1. A probability of 0 means the event never happens, 1 means it always happens, and 0.5 means it is equally likely to happen or not.

Q: What does 'equally likely' mean?

It is the assumption that every outcome occurs with the same probability. It is justified when there is physical or structural symmetry — the six faces of a die, or the two sides of a coin — that gives no outcome an advantage. The basic formula 'favourable / total' is valid only under this assumption.

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 actually happens.

Probability is, in short, a way of putting a number on uncertainty.

Figure 1.1: "Probability" in everyday life

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 most basic idea behind probability.

Basic formula for probability (when outcomes are equally likely)

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

Note: This formula is valid only when every outcome can be assumed "equally likely." See Section 1.5 for details.

Figure 1.2: The probability of rolling a 1 on a die

1.3 Why probabilities lie between 0 and 1

Why is a probability always between 0 and 1?

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

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).

Figure 1.3: Probabilities range from 0 to 1

Probability	Meaning
0	Never happens
0.5	Equally likely to happen or not (50/50)
1	Always happens

1.4 Notation for probability

In mathematics, probability is written as:

$$P(A)$$

This means "the probability that A occurs." P stands for Probability; A denotes the event of interest (called an event).

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?

A die is a cube; every face has the same shape and area.
Its centre of mass is at the centre, with no bias.
So physically, no face is favoured over another.

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

"Equally likely" = every outcome occurs with the same probability.
This is more than a paraphrase: it is the mathematical assumption that, by reason of symmetry (or some equivalent argument), all outcomes carry the same probability.

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.

Figure 1.4: Are the outcomes equally likely?

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?

$$1000 \times \dfrac{1}{6} \approx 167 \text{ times}$$

Experimentally, the actual count clusters around this value.

Probability captures 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.

Figure 1.5: As the number of trials grows, the proportion approaches the probability

1.7 Chapter Summary

Concept	Meaning
Probability	A way of expressing uncertainty as a number between 0 and 1
Basic formula	favourable outcomes ÷ total outcomes
Equally likely	Every outcome occurs with the same probability
Frequentist interpretation	Long-run proportion under repeated trials

Exercises

Problem 1

What is the probability of getting heads when tossing a coin once?

Solution to Problem 1

$\dfrac{1}{2}$ (one of two outcomes: heads or tails).

Problem 2

What is the probability of drawing a heart from a standard deck of 52 cards?

Solution to Problem 2

$\dfrac{13}{52} = \dfrac{1}{4}$ (13 hearts out of 52 cards).

Problem 3

You pick one integer at random from 1 to 10. What is the probability that it is even?

Solution to Problem 3

$\dfrac{5}{10} = \dfrac{1}{2}$ (the even numbers are 2, 4, 6, 8, 10 — five values).