6. Probability of an Event
Definition of a Probability
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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
The next 2 sections give more examples of probability:
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