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
On this page...
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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.
Thus,
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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
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:

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
The next 2 sections give more examples of probability:
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