Probability and sample spaces

Random experiments, equally likely outcomes, the formula P(E) = |E|/|S|, playing cards, impossible and certain events, complements, invalid arguments and relative frequency.

The basic formula

A random experiment has a sample space $S$ of possible outcomes. An event $E$ is a set of outcomes. If all outcomes are equally likely:

Probability

$$P(E) = \frac{|E|}{|S|} = \frac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}$$

$0 \le P(E) \le 1$. An impossible event has probability $0$; a certain event has probability $1$.
Complement: $P(\overline{E}) = 1 - P(E)$, the probability that $E$ does not happen.

Playing cards

A standard pack has $52$ cards in four suits (hearts and diamonds are red; clubs and spades are black). Each suit has $13$ cards: ace, $2$ to $10$, and the picture cards jack, queen and king.

Example 1●

A card is drawn. Find the probability that it is a king, and that it is not a picture card.

$P(\text{king}) = \dfrac{4}{52} = \dfrac{1}{13}$

$P(\text{picture}) = \dfrac{12}{52}$, so $P(\text{not picture}) = \dfrac{40}{52} = \dfrac{10}{13}$.

There are 3 picture cards in each of 4 suits.

Relative frequency

When outcomes are not equally likely, estimate probability by experiment: $\text{relative frequency} = \dfrac{\text{number of times the event occurred}}{\text{number of trials}}$. More trials give a better estimate.

Flashcards

$P(E)$ for equally likely outcomes

$\dfrac{|E|}{|S|}$

$P(overline{E})$

$1 - P(E)$

Cards in a pack

$52$: $4$ suits of $13$, $12$ picture cards

Relative frequency

Occurrences $\div$ trials

Practice

Find the probability of rolling a $5$ on a fair die.
Find the probability of drawing a king from a pack of $52$.
Find the probability of drawing a heart.
Find the probability that a card is not a picture card.
An event with probability $0$ is:
A coin is tossed $200$ times and lands heads $112$ times. Find the relative frequency of heads.
Find the probability of rolling a prime number on a die.
"A seed either sprouts or it doesn't, so $P(\text{sprouts}) = \tfrac12$." This is:
Find the probability of drawing a red ace.