Events

The event is one of the keywords that we come across often in the study of probability. The study begins with the introduction of probability or random experiments. A probability/random experiment results in well defined, uncertain and distinct outcomes. The set of all possible such outcomes constitute the sample space.

Event Definition

An event consists of a set of outcomes of a probability experiment and thus a subset of sample space.

Types of Events

We come across many types of events in the study of both theoretical and empirical probabilities. The general probability laws get modified according the event type in discussion. The events are broadly classified into two types on the basis of number of outcomes.
      1.    Simple Event
      2.    compound Event

Simple Event

If the sample space has n equally likely outcomes, an event with only one such outcome is called a simple event.

Thus each outcome in the sample space can be viewed as a simple event. The classical or theoretical probability is defined on the assumption of equal chance of these simple events occurring. A simple event is also called an elementary event.

Examples for simple events:

1. In the simple experiment of throwing a coin, the two likely events, turning a head and turning a tail are simple events, as these have only one outcome in their respective sets

                                                                          Sample space  =  {H, T}                                                                 
                  Turning a Head = {H}                Turning a tail = {T}

2.  In the experiment  of rolling a six faced die, the six distinct and equally likely events of getting 1, getting 2,etc are simple events

                                                                    Sample Space = {1,2,3,4,5,6}
      Rolling 1 = {1}          Rolling 2 = {2}          Rolling 3 = {3}        
     Rolling 4 = {4}       
        Rolling 5 = {5}           
        Rolling 6 = {6}           

Compound Event

Compound events are defined by combining simple events.

In the above two examples of tossing a coin and throwing a die the compound events can be defined as follows
     1. Turning a head or tail = {H, T}
     2. Throwing and even number = {2, 4, 6}
Broadly there are two methods of compounding events.
     1. Union of Events
     2. Intersection of Events

Union of Events

A compound event is called a union of two events, if its elements are obtained by considering the elements of either of the two events or both the events.

The union of two events A and B is the set of outcomes that are included in A or B or in both.

The union of A and B is represented by AUB

In the adjoining Venn diagram the shaded region showing all of A and B represents AUB.

Union Of Events

While counting the elements or outcomes for the union, the common outcomes are counted only once.

Example: 

In a class of 40, students need to take two elective subjects. 18 students had Math as one of the electives, 16 had Statistics as one of the chosen subject.  8 students had chosen both statistics and Math as their preferred subjects.  If a student is chosen at random what is the probability that he has taken Math, Stat or both as his electives.

The probability of an event = $\frac{Number \ of \ outcomes \ favorable \ to \ the \ event}{Number \ of \ all \ possible  \ outcomes}$

When we add the numbers given for the two subjects, we are counting the number of students who have taken both the subjects twice.  So to find the number of elements in the union, we have to subtract the number of students who have taken both from this sum.

The number of students taken Math, Statistics or both = 18 + 16 - 8 = 26

Probability the selected student had taken either Math or Statistics or both = $\frac{26}{40}$ = 0.65

Intersection of Events

The compound event obtained by considering the outcomes common to two events is known as the intersection of events.

The intersection of two events A and B is the set of outcomes common to both the events.

The intersection A and B is represented by A ∩ B.

In the adjoining Venn diagram the shaded region represents the portion common to both the events A and B

Intersection Of Events
Based on the counting of number of elements in the sets, their union and intersection, the probabilities of the two compound events are related by the following formula.

Addition formula for probabilities

P(A U B) = P(A) + P(B) - P(A ∩ B)

Mutually Exclusive Events

Two events A and B are said to be mutually exclusive if they do not have common outcomes. This means they cannot occur simultaneously.

Two or more events are called mutually exclusive events if the occurrence of one event denies the occurrence of the others. These events are also called disjoint events.
When two events A and B are mutually exclusive, then the compound event A ∩ B cannot take place. This means P(A ∩ B) = 0. The addition rule for probability for mutually exclusive events are therefore,

                                                      P(A ∪ B) = P(A) + P(B)

Example:

There are 12 boys and 10 girls in a class. A student is selected as the class representative. If B is the event of selecting a boy and G the event of selecting a girl as the representative, then the two events B and G are mutually exclusive.

Exhaustive Events

If E1, E2, E3…….En are mutually exclusive events and if the sample space does not contain any other events, the above said events are called mutually exclusive and exhaustive events. The addition rule for probabilities for mutually exclusive and exhaustive events is as follows

                               P(E1) + P(E2) + P(E3) + …………P(En)   = 1

Example for exhaustive events

When a six faced die is thrown the events of throwing 1, throwing 2, throwing 3,throwing 4, throwing 5 or throwing 6 are mutually exclusive and exhaustive events.

These six events are equally likely events and the probability of each event = $\frac{1}{6}$

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ + $\frac{1}{6}$ = 1

Independent Events

Two events are said to be independent if they can occur simultaneously. This means the occurrence of one does not depend on the occurrence or non occurrence of the other event. For example if a die is thrown and simultaneously one card is picked from a pack, then the events of getting a six and picking an ace are independent events.

In the case of mutually exclusive events, the occurrence of one event depends on the non occurrence of other events. Hence mutually exclusive events are dependent events.

The multiplication rule for the conditional probability states

P(A | B) = $\frac{P(A\bigcap B)}{P(B)}$ where P(A | B) is the conditional probability of event A when it is known that the event B has already occurred.
In the case of A and B being independent events, P(A | B) reduces to P(A).

Hence the multiplication rule for independent events A and B is stated as

                                                 P (A ∩ B) = P(A) . P(B)