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Probability

Events and Its Algebra

Probability Definition: Suppose you are playing with some of your friends. In this game, you have to throw a die and getting six is considered as lucky. The more six you get, the more is the chance of your winning. How can you calculate the chance of winning? Is the chance of winning the same for all the friends? Here getting a six is an event.

Are all the events the same for all? Since now we are familiar with probability definition. We are able to answer probability formulas. In this section, we will study about events, the various types of events and its algebra and probability definition in short.

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Probability Definition

The probability of any event is defined as the chance of occurrence of the events to the total possible outcomes. If there are ‘n’ exhaustive, mutually exclusive and equally likely outcomes of a random experiment. Out of which, ‘m’ are favorable to the occurrence of an event E.

The probability definition is given as the ratio of the number of favorable events to the total number of exhaustive ones. This probability is equal to m⁄n.

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Events and Its Algebra

Any subset of a sample space is an event. In other words, a combination of outcomes of a random experiment is an event. It is denoted by capital letters.

In a random experiment of throwing a die, an event can be of getting any of the numbers from 1 to 6 on its uppermost face. We can calculate the probability of any of the possible events. For example, the probability of an event of getting 5 in a single throw of a die is 1⁄6.

Types of Events

Based on random experiments, events are of following types.

Impossible Events

The events which are impossible to happen come in the category of impossible events. The empty set Φ is an impossible set. Consider an example of a pack of 52 playing cards, the event of getting a card of number 12 is impossible.

Sure Events

The set of all possible outcomes which are certain to occur form sure event. The entire sample space is a sure event. For example, in a random experiment of tossing of a coin the event of getting a tail is a sure event.

Learn the Probability of an Event here. 

Simple Event

Any event is Simple if it corresponds to a single possible outcome of the experiment. In other words, if any event has only one sample point of a sample space it is a simple event. In a random experiment of throwing of a die, the sample space

S = {1, 2, 3, 4, 5, 6}. The event, E, of getting 5 on the uppermost face is a simple event.

Compound Event

Any event is Compound if it corresponds to more than a single possible outcome of the experiment. In other words, if any event has more than one sample point of a sample space it is a compound event. In a random experiment of throwing of a die, the sample space

S = {1, 2, 3, 4, 5, 6}. The event E of getting a multiple of 2 is a compound event as E = {2, 4, 6}.

Based on the Set theory, we can perform some algebra on events. Some of them are the union or the intersection of events. Let us study them in details.

Complimentary Event

As the name suggests, complementary of any event shows its contrary side. For any event E, the complementary event E’ shows not E. Every outcome which is not in E, can be assumed to be in E’. In simple terms, if E denotes the glass is half empty, the event E’ shows that the glass is half filled.

Let us take an example of throwing a die. The sample space, S = {1, 2, 3, 4, 5, 6}. E shows the event of getting even number i.e., E = {2, 4, 6}. The event E’ shows the outcome of not an even number or getting an odd number. E’ = {1, 3, 5}.

Event A or B

The event A or B shows the sample points of a random experiment which are either in A or B or both. Event A or B = A ∪ B Suppose event A = {1, 3, 4, 7} and B = {2, 3, 5. 6}. A ∪ B = {1, 2, 3, 4, 5, 6, 7}.

Event A and B

Let A and B be two events. The event A and B show the sample points of a random experiment which are common to both A and B. It is similar to the intersection of two sets A and B. Event A and B = A ∩ B.

Consider a random experiment of throwing a die. A is the event of getting an even number. B is the event of getting a multiple of 3. A ∩ B shows the sample point which is common to both A and B. Here, A = {2, 4, 6} and B = {3, 6} and A ∩ B = {6}.

Event A but not B

The event A but not B shows the sample points which are in A but not in B. Event A but not B = A ∩ B’ = A – A ∩ B. This event shows the unique sample points of A other than that in B. Suppose event A = {1, 3, 4, 5, 6, 7} and B’ = {2, 3, 5. 6}. A ∩ B’ = {1, 4, 7}.

Events Bases on Venn Diagram

A Venn diagram showing various algebra of events.

probability definition

Below are some of the used statements and their meaning in terms of set theory.

Statement Meaning in terms of Set Theory
Sample Space Universal Set, S
Complimentary Event of A A’ or Ac or A‾
At least one of the events A or B occurs A ∪ B
Both the events A and B occurs A ∩ B
Events A and B are mutually exclusive A ∩ B = Φ
Neither A nor B occurs A’ ∩ B’
Event A occurs and B does not occur A ∩ B’
Exactly one of the events A or B occurs (A ∩ B’) ∪ (A’ ∩ B)
Not more than one of the events A or B occurs (A ∩ B’) ∪ (A’ ∩ B) ∪ (A’ ∩ B’)
If event A occurs, so does B A ⊂ B

Based on the above algebra of events, we can define some other events.

Exhaustive Events

The total number of possible outcomes of a random experiment is an exhaustive event. The event of getting an odd number and an event of getting an even number in a throw of a die together forms an exhaustive event.

Favorable Events

The numbers of outcomes which show the happening of the event in a random experiment are favorable events. They show the number of cases favorable to an event. In a random experiment of tossing of two coins, the number of the favorable event for getting two tails together is 1.

Mutually Exclusive Events

Events are mutually exclusive if the happening of any one of the events excludes the chance of happening of the other in the same trail. Also, we can say that no two or more of the events can happen simultaneously. The events of head and tail in tossing a coin are mutually exclusive. Only one of them can happen.

Equally Likely Events

An event in which all the outcome has an equal chance to occur. In throwing a die, all six faces are equally likely to come.

Independent Events

Two events are independent if there is no effect on the happening and the non-happening of one by the others. In throwing a die, the result of getting 2 in a first throw does not affect the result of second and other throws.

The Probability Definition of an Event

Below is the list of probability definition associated with an event.

Probability of event ‘A or B’ = P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
Probability of event ‘not A’  = P(A’) = 1 – P(A)
The probability of event ‘A but not B’ = P(A ∩ B’) = P(A) – P(A ∩ B)
A probability of event ‘not A not B’ = P(A’ ∩ B’) = 1 – P(A ∪ B) = 1 – Probability of event A or B.

Solved Examples for You

Question 1: Problem: Suppose there are 4 red, 6 blue and 2 green balls in a bag. Two balls are drawn at random from the bag. Find the probability of that the two balls drawn are red.

Answer: Total number of balls in the bag = 4 + 6 + 2 = 12. Two balls are drawn at random. The total number of ways in which any two balls are drawn = 12C2 = 66. The number of favorable cases of getting two red balls = 4C2 = 6. Therefore, the required probability =  6⁄66 = 1/11.

Question 2: Problem: A, B, and C are three mutually exclusive and exhaustive events of a random experiment. If P(A) = 4⁄5 P(C) and P(B) = 3⁄5 P(C). Find P(C).

Answer: Since A, B and C are mutually exclusive and exhaustive events, P(A) + P(B) + P(C) = 1.
⇒ 4⁄5 P(C) + 3⁄5 P(C) + P(C) = 12⁄5 P(C) = 1 or, P(C) = 5⁄12.

Question 3: What is meant by probability?

Answer: Probability refers to the number of ways of achieving success. Furthermore, probability also refers to the total number of possible outcomes. For example, the probability of getting a head after flipping a coin is ½, because there is only 1 way of getting a head while the total number of possible outcomes happens to be 2. So, one must write P(heads) = ½.

Question 4: Explain the significance of probability?

Answer: Probability helps in finding out the maximum percentage of occurrence of any event. Probability will enable one to know how likely it is that an event can occur. Probability helps in understanding the prediction of happening. It provides a rough idea regarding an outcome’s occurrence.

Question 5: Explain the product rule of probability?

Answer: The product rule of probability states that the probability of two or more independent events whose occurrence takes place together can be calculated by doing the multiplication of the individual probabilities of the events.

Question 6: How many types of probability are in existence?

Answer: There are four types of probabilities in existence. These four types are classic probability, experimental probability, theoretical probability, and subjective probability.

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