Probability Basics

≈ 20 min

Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). For equally likely outcomes,

P(E) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.

The set of all outcomes is the sample space. Every probability obeys $0 \le P(E) \le 1$ , and the probabilities of all outcomes sum to 1.

The complement $E'$ is ' $E$ does not happen', and

P(E') = 1 - P(E).

This is often the quickest route: to find $P(\text{at least one})$ , subtract $P(\text{none})$ from 1.

Worked example. A fair die has sample space $\{1,2,3,4,5,6\}$ . The event 'even' is $\{2,4,6\}$ , so $P(\text{even}) = \dfrac{3}{6} = 0.5$ .

Which of these values cannot be a probability?

$0$

$0.5$

$1$

$1.3$

A fair six-sided die is rolled. What is the probability of getting an even number? Give a decimal.

A bag holds 5 red and 3 blue marbles. Drawing one at random, what is $P(\text{red})$ ? Give a decimal.

Two fair dice are rolled. What is the probability that the two numbers sum to 7? Give a decimal to about 4 places.