Theoretical probability

Sample spaces, equally likely outcomes, calculating theoretical probability, and the probability of an event not happening (the complement).

Sample space

The sample space is the list of all possible outcomes. Rolling a die: $\{1, 2, 3, 4, 5, 6\}$. An event is a set of outcomes, such as "rolling an even number" $= \{2, 4, 6\}$.

$$P(A) = \frac{\text{number of outcomes in } A}{\text{total number of outcomes}}$$

Example 1●

A die is rolled. Find $P(\text{a number less than } 3)$.

Favourable: $\{1, 2\}$, so $2$ outcomes out of $6$.

$P = \dfrac{2}{6} = \dfrac{1}{3}$

The complement

Example 2●●

The letters of MATHEMATICS are on cards. One is picked at random. Find (a) $P(\text{M})$ and (b) $P(\text{not a vowel})$.

There are $11$ letters, with $2$ Ms. (a) $P(\text{M}) = \dfrac{2}{11}$.

Vowels: A, E, A, I, so $4$ vowels. (b) $P(\text{not a vowel}) = 1 - \dfrac{4}{11} = \dfrac{7}{11}$.

Flashcards

Sample space

The list of all possible outcomes.

$P( ext{not } A)$

$1 - P(A)$

$P( ext{even})$ on a die

$\tfrac{3}{6} = \tfrac{1}{2}$

Practice

A die is rolled. Find $P(\text{less than } 3)$.
A letter of MATHEMATICS is chosen. Find $P(\text{M})$.
A letter of MATHEMATICS is chosen. Find $P(\text{not a vowel})$.
$P(\text{rain}) = 0.3$. Find $P(\text{no rain})$.
A card is drawn from a standard deck of $52$. Find $P(\text{heart})$.
A card is drawn from a standard deck. Find $P(\text{ace})$.
A spinner has numbers $1$ to $10$. Find $P(\text{a multiple of } 3)$.
How many outcomes are in the sample space for one die roll?