Types of Events in Probability

Probability can be simply defined as the likelihood of a particular event to take place. Probability generally lies between 0 and 1. The 0 indicates impossibility of the event and on the other hand, 1 indicates surity of a particular event. Probability can be defined as the percentage of a particular event. The only difference is it cannot be 0% or 100%. This is because of the fact that 0% means the particular event is impossible whereas 100% means surity of a particular event.

What are Events?

The result or the outcome of a random experiment are called events connected with the experiments. Thus, "head" and "tail" are the results of the random experiment of tossing an unbiased coin and are events connected with this experiment. Similarly, the outcomes of the random experiment of throwing an unbiased die from a box are 1, 2, 3, ....., 6 and each outcome is an event connected with this experiment. Similarly, the outcome is an event connected with this experiment. Usually, capital letters A, B, C, etc are used to denote events connected with a random experiment. 

The different types of events of probability are described in details below:

1) Compound or Composite Event

Let A and B be the events 'even face' and 'multiple of three' respectively in the random experiment of throwing an unbiased die. Clearly, the event A occurs when the result of the experiment is 'two' or 'four' or 'six': similarly, the event B occurs if the outcome of the experiment is 'three' or 'six'. Thus, event A can be decomposed into the events 'two', 'four', 'six'. Thus event A can be 'three' and 'six'. Such events are called compound events or Composite events. 

In simple words, Unlike simple events, while considering sample space, when an event consists of more than one event, then the event is termed as a compound event. 

2) Simple or Elementary Event

On the contrary, events 'one', 'two', etc connected with the same experiments cannot be composed further: such events are called Simple or Elementary events. Thus, decomposed, whereas the compound event is the outcome of a random experiment that can be decomposed further into simple events.

In a sample space, when an event consists of a single point, then the event is said to be a simple event .

In order to understand it more elaborately, let us consider the example

Suppose, S = {56, 78, 96, 89, 54}

and E= {78}. So, E is a simple event.

3) Mutually Exclusive Events

Two events A and B connected with a random experiment E, are said to be mutually exclusive if they cannot occur simultaneously. Symbolically events A and B are mutually exclusive when A union B is equal to null or impossible events. Two simple events connected with a random experiment are always mutually exclusive but two compound events may or may not be so. Let A, B, and C be the events 'even face', 'odd face', and 'a multiple of three' respectively in the random experiment throwing an unbiased die. Clearly, the events A and B cannot occur simultaneously, and hence, they are mutually exclusive: but the events B and C occur simultaneously if the result of the experiment is three and hence, they are not mutually exclusive.

If two events don't have any common point between each other, then the event is said to be mutually exclusive. This can be defined as " If the occurrence of one event excludes the occurrence of the another event, then the event is termed a mutually exclusive event.

4) Impossible and Certain (or sue) Events

In any random experiment we can think of an event that is logically impossible i.e., cannot occur at any performance of the experiment. Such an event is called an impossible event. For example, the event of drawing a black ball from a bag containing three red and four white balls is an impossible event. Similarly, the event 'seven' can never occur in the random experiment of throwing a die and is, therefore, an impossible event. An impossible event is denoted by f and the probability of occurrence of an impossible event is zero. Again, in any random experiment, we can think of an event that is sure to occur at every performance of the experiment. Such an event will be called a certain (or sure) event. Let A denote the event 'Head or Tail' in the random experiment of tossing an unbiased coin. Clearly, event A is sure to occur at every performance of the experiment and is, therefore, a certain event. A certain event is usually denoted by S and the probability of occurrence of a certain event is one.

In simple terms, When the probability of a particular event turns out to be 0, then it is termed as an impossible event. On the other hand, when the probability of a particular event turns out to be 1, then the event of the probability is termed as a sure event.

5) Equally Likely Events

Two or more events are said to be equally likely if, after taking into consideration all relevant evidence, none can be expected in preference to another. Simple events connected with a random experiment are always equally likely but compound events may or may not be so. Thus the simple events 'Head' and 'Tail' connected with the random experiment of tossing an unbiased coin are equally likely. Similarly, the six simple events 'one', 'two', ...., 'six' connected with the random experiment of throwing an unbiased die are equally likely but the compound events A and B are equally likely when P(A) = P(B).

Independent Event and Dependent Event

When a occurence of any event is dependent on the occurrence of the other event, then the event is known as dependent event. On the other hand, if an occurence of any event is independent of the occurrence of the other event, then the event is known as independent event.

Complementary Events

In order to understand the particular type of event, let us consider a certain example. 

Suppose, for any sample, S, E₁ is the event and E₂' is the remaining elements of the sample. 

So, this can be simply written as,

E₁= S - E₁'

Let us suppose that a dice is rolled.

Then the sample space S ={1,2,3,4,5,6}.

Now, let us consider that , E1 represents all the outcomes greater than 4. Then, 

E₁= {5,6} and E₁' = {1,2,3,4}.

So, taking into consideration all the instances stated above, we can assume that E₁' is a complement of the event E₁. 

Thus, we can say that , the complement of the event E₁,E₂,E₃,........E_n will have a complementary E₁,E₂,E₃,........E_n.

Events Associated with "OR"

Let us consider that E₁ and E₂' are associated with OR, then it implies that either E₁ or E₂ or both. The symbol(U) represents the OR, which denotes union. 

Thus the following instances can be written as E₁ U E₂.

Let us consider E₁,E₂,E₃,........E_n, are mutually exhaustive events which are associated with the sample space S, then it can be written as E₁ U E₂ U E₃ U ……...E_n.

Event E1 But Not E2

When a particular type of event represents the difference between two events. Then the event is known as Event E₁ but not E₂. The outcomes of E₁ are not present in the outcomes of E₂. So, This can be simply represented as:

E₁,E₂ = E₁ - E₂

Events Associated with 'AND'

Two events when associated with an "And", implies that there is an intersection of elements which is common to each other. 

Probability is Used in Different Places Which Includes

Weather

Meteorologists use the probability in order to predict how likely an event has to occur which includes snow, rain or other weather. They cannot tell the exact weather conditions , so they use instruments or tools which tell them a particular percentage of the event to occur.

Sports

Probability, in this case, helps to predict the probability of winning or losing of the particular teams. These are generally used by coaches or athletes.

Insurance

These are used in order to predict how likely a particular car or any other thing would get insurance. 

Games

During playing card games, board games or video games, probability is used. Often the odds of the events are measured in order to predict the likelihood of winning.

Solved Examples

Question 1: Two unbiased dice are rolled together. Find the odds in favor of getting 2 digits, the sum of which is 7. 

Solution: Evidently the first die may have 6 different outcomes, each of which can be associated with 6 different outcomes of the second die. Therefore, the sample space of the random experiment of throwing two unbiased dice together contains 6 x 6 = 36 equally likely event points. Let A denote the event that the sum of the digits in the two dice is 7. Clearly, event A contains 6 equally likely events viz., (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).Therefore, by the classical definition of probability, we get, P(A) = 6/36 = ⅙Therefore, odds in favor of events A are 1:(6-1) = 1:5

Question 2: the odds in favor of an event are 4:3. The odds against another independent event are 2:3. What is the probability that at least one of the events will occur? 

Solution: Assume that the given events are A and B.Then by the problem, probability of occurrence of A = P(A) = \[ \frac{4}{(4+3)} = \frac{4}{7} \]

And probability of occurrence of  B = P(B) = \[ \frac{3}{(2+3)} = \frac{3}{5} \]

Therefore, the probability of occurrence of at least one of the events A and B= P(A⋃B) = P(A) + P(B) - P(A⋂B)

= P(A) + P(B) - P(A).P(B)=  \[ \frac{4}{7} + \frac{3}{5}-\frac{4}{7} . \frac{3}{5} = \frac{(20+21-12)}{35} = \frac{29}{35} \].