PROBABILITY

PROBABILITY

Elements of a Probability

Sample Spaces and Events
The set of all possible outcomes is called the sample space of the experiment, and is denoted by ?.
Ex. Rolling a die sample space: S = {1,2,3,4,5,6}
And ,Tossing a coin: S = {H,T}
Ex: Rolling a die. The experiment is rolling a die. A common die is a small cube whose faces shows numbers 1, 2, 3, 4, 5, 6 one way or another. These may be the real digits or arrangements of an appropriate number of dots, e.g. like these

There are six possible outcomes and the sample space consists of six elements:
{1, 2, 3, 4, 5, 6}.

An outcome is a result of a some activity.

Example. Rolling a die has six outcomes: 1, 2, 3, 4, 5, 6.

A subset of the sample space, that is, a collection of possible outcomes, is called an event.
The sample space of an experiment may consist of a finite or an infinite number of possible outcomes.
Throwing a die.
Sample space:
S = {1, 2, 3, 4, 5, 6} or S = {Even, odd}
Some events:
Even numbers, E1 = {2, 4, 6}
Odd numbers, E2 = {1, 3, 5}
The number 1 , E3 = {1}
At least 3, E4 = {3, 4, 5, 6}
Operations with Events: Intersection, Union, Complement

We now consider operations with events that will result in the formation of new events – these new events will be subsets of the same sample space as the given events.

Definitions

The Intersection of two events A and B, denoted by the symbol A?B, is the event containing all the sample points that are common to A and B.
The Union of two event A and B, denoted by the symbol A?B, is the event containing all the sample points that belong to A or B or both.
The Complement of an event A with respect to a sample space S is the set of all sample points of S that are not in A. The complement of A is denoted by
DeMorgan’s laws. For any two events A and B we have

,




mutually exclusive: there are some events that can never occur together.
For example, it is impossible that a coin can come up both heads and tails.
And it is impossible that a steel pin can be both too long and too short.
The events A and B are said to be mutually exclusive if they have no outcome in common.
Equally likely outcomes
If the experiment is of such a nature that we can assume equal weights for the sample points of S, i.e “Have same probability to occur”

then the probability of any event A is the ratio of the number of elements in A to the number of elements in S.
P(A)=n(A)/n(S)