6. Probability of an Event

Definition of a Probability

Suppose an event E can happen in r ways out of a total of n possible equally likely ways.

Then the probability of occurrence of the event (called its success) is denoted by


The probability of non-occurrence of the event (called its failure) is denoted by


Notice the bar above the E, indicating the event does not occur.



In words, this means that the sum of the probabilities in any experiment is `1`.

Definition of Probability using Sample Spaces

When an experiment is performed, we set up a sample space of all possible outcomes.

In a sample of N equally likely outcomes we assign a chance (or weight) of `1/N` to each outcome.

We define the probability of an event for such a sample as follows:

The probability of an event E is defined as the number of outcomes favourable to E divided by the total number of equally likely outcomes in the sample space S of the experiment.

That is:



Properties of Probability

(a) 0 ≤ P(event) ≤ 1

In words, this means that the probability of an event must be a number between `0` and `1` (inclusive).

(b) P(impossible event) = 0

In words: The probability of an impossible event is `0`.

(c) P(certain event) = 1

In words: The probability of an absolutely certain event is `1`.

Example 1

What is the probability of...

(a) Getting an ace if I choose a card at random from a standard pack of `52` playing cards.

(b) Getting a `5` if I roll a die.

(c) Getting an even number if I roll a die.

(d) Having one Tuesday in this week?

Example 2

There are `15` balls numbered `1` to `15`, in a bag. If a person selects one at random, what is the probability that the number printed on the ball will be a prime number greater than `5`?

Example 3

The names of four directors of a company will be placed in a hat and a 2-member delegation will be selected at random to represent the company at an international meeting. Let A, B, C and D denote the directors of the company. What is the probability that

(a) A is selected? (b) A or B is selected? (c) A is not selected?

Coming next...

♥ 2 3 4 5 6 7 8 9 10 J Q K A
♦ 2 3 4 5 6 7 8 9 10 J Q K A
♣ 2 3 4 5 6 7 8 9 10 J Q K A
♠ 2 3 4 5 6 7 8 9 10 J Q K A

The next 2 sections give more examples of probability:

Singapore Toto


Didn't find what you are looking for on this page? Try search:

Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Ready for a break?


Play a math game.

(Well, not really a math game, but each game was made using math...)

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

Given name: * required

Family name:

email: * required

See the Interactive Mathematics spam guarantee.

Share IntMath!

Short URL for this Page

Save typing! You can use this URL to reach this page:


Math Lessons on DVD


Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos