# 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

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