Definition of Probability

3. Probability: Probability

Definition of Probability

In everyday language, probabilities are often described as percentages. In statistics, however, all probabilities are expressed as a number between $0$ and $1$ .

"The probability that a fair coin comes up head is $50\%$ " translates to "the probability that a fair coin comes up head is $0.5$ " in statistics.

A probability of $0$ or a probability of $1$ follow these rules:

An event that has no chance of occurring has a probability of
- e.g. the probability that someone is born on February 30th is $0$ .
An event that is certain to occur has a probability of
- e.g. if today is Thursday, then the probability that tomorrow is Friday is $1$ .

The greater the probability of an event, the more likely it is that the event will occur.

Probability

Definition

Probability is the likelihood that an event will occur, quantified as a number between $0$ and $1$ .

If the probability of an event $A$ occurring is $p$ , this is written as: $\mathbb{P}(A)=p$ .

For an experiment in which several different outcomes are possible, the probability of any specific outcome is defined as a fraction or proportion of all the possible outcomes.

Formula

$\mathbb{P}(A) = \cfrac{\text{number of outcomes classified as }A}{\text{total number of possible outcomes}}$

Rules

The probability of any event A, $\mathbb{P}(A)$ , is always greater than or equal to $0$ and less than
or equal to $1$ :
$0\leq \mathbb{P}(A) \leq 1$
The sum of the probabilities of all possible outcomes must equal $1$ . That is, if the sample space is $\Omega$ , then
$\sum_{\text{all }x \text{ in }\Omega}\mathbb{P}(x)=1$

Consider the random experiment of a single coin toss with sample space $\Omega = \{H,T\}$ .

Event $A$ is defined as $A =$ 'you get heads' = $\{H\}$ .

What is the probability of event $A$ happening?

$\mathbb{P}(A)=\cfrac{\text{number of outcomes classified as 'you get heads'}}{\text{total number of possible outcomes}}=\cfrac{1}{2}=0.5$

New example