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.
`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`.
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?
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`?
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?
♦ 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: