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

`P(E)=r/n`

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

`P(barE)=(n-r)/n=1-r/n`

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

Thus,

`P(barE)+P(E)=1`

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:

`P(E)=(n(E))/(n(S)`

where

  • `n(E)` is the number of outcomes favourable to E and

  • `n(S)` is the total number of equally likely outcomes in the sample space S of the experiment.

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

Poker