Any subset of a sample space is called an event. If an event has one element only it is then called a simple event. This is the formal definition of the word “event” we use in the theory of probability; it is different from what we mean by this word in everyday language.

To understand the concept of an “event” in the context of probability, let us try an example. Assume that two dice are rolled together. Then the sample space, S, of this experiment is as follows:

Any subset of this set is called an event of rolling two dice. For example, the sets {(1 1), (2 2), (3 3), (4 4), (5 5), (6 6)} and {(1 2), (2 4), (3 6)} are events that can be observed as the result of the experiment. These events are highlighted in different colors in the chart below:

Probability of an Event in a Uniform Sample Space

1.  If an event E results in n(E) equally likely outcomes and the sample space S has n(S) equally likely outcomes, then the probability of occurring the event E is
   .

2. The number of the outcomes of an event must be less than or equal to the number of the outcomes in the sample space. Thus
   .

3. The closer a probability of an event is to 1 the highly probable it occurs.

4. The probability of an event can be presented as a fraction, decimal, or a percent.

Practice 1. Tumblin’s Blocks
In the figure below, a set of Tumblin’s blocks shown. If two circular boards thrown toward the set and land on the horizontal squares, what is the probability of getting a total less than 12?

Solution
The sample experiment is

Let E be the set of all outcomes in which the total of the two numbers is less than 12. Then

Now let's do Practice Exercise 7-1-a

Impossible Event
If an event is an empty set, then it is called the impossible event. For example, in rolling two dice, if our expectation is to get two numbers whose sum is less than 2, then such a set can not be established using the set S; since in S, the sum of numbers in each pair is equal to or greater than 2. The probability of such an event is denoted by P(E) = 0.

Certainty Event
If an event is the same as the sample space then it is called a certainty event. For example, in rolling two dice, the sample space S is called the certainty event. The probability of such an event is denoted by P(E) = 1.

Practice 2. Two dice rolled together, what is the probability of getting the same numbers on both dice?

Solution
The sample space of this experiment has 36 elements. So, we assign the probability of to each outcome. The event we expect to occur has the following elements from the sample space, which contains six elements: {(1 1), (2 2), (3 3), (4 4), (5 5), (6 6)}. Thus, the probability of getting the same numbers on both dice is .

How to Find the Number of Events of a Sample Space?
If a sample space has n elements, then the number of different events can be observed is equal to 2ⁿ. For example, if a sample space has 5 elements, then the number of events for this experiment is 2⁵ = 32.

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