Events in Probability – Types, Definitions, Questions | Independent, Sample, Mutually Exclusive Events

The probability of an event is the set of outcomes of an experiment. In other words, events in probability are the subset of the respective sample space. The possible set of outcomes of a random experiment is called the sample space. The chance of occurrence of an event is the probability. Interested students can check the below sections to know what is an event, its types and examples.

What is an Event?

The set of outcomes from an experiment is called an event. For example, when you toss a coin. The outcome of this experiment is head or tail. These are the events connected with an experiment. In any experiment, there is a probability that is either an event occurs or not. The probability of occurrence of an event lies between 0 and 1.

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Types of Events in Probability and Definitions

Following are the types of events in probability.

1. Sample Event

If an event E has only one sample point of a sample space, it is called a sample event or elementary event. It is an event that contains exactly one outcome.

Example: Throw a die, the possibility of appearing 3 on the die is a sample event and it is E = {3}.

2. Compound Event

It is opposite to the sample event. If there is more than one sample point on the sample space, then such an event is called the compound event. It involves the combination of two or more events together and finding the probability of a combination of events.

Example: When you throw a die, the possibility of an odd number appearing is a compound event, as there is more than one possibility, there are three possibilities i.e E = {1, 3, 5}.

3. Impossible Event

When an event cannot occur that means there is no chance of occurring an event is called an impossible event. The probability of this type of event is 0.

Example: The probability that the card is drawn from a deck is both red and black is an impossible event.

4. Equally likely Events

If the outcomes of an experiment are equally likely to happen, then they are called equally likely events. Like during a coin toss you are equally likely to get heads or tails.

5. Mutually Exclusive Events

Two events are said to be mutually exclusive events when both events cannot occur at the same time. Mutually exclusive events always have a different outcome. Two simple events are always mutually exclusive, whereas two compound events may or maybe

If A and B are two events, then

A ∩ B = Ø

P(A ∩ B ) = 0

P(A U B) = P(A) + P(B)

6. Complementary Events

For event E, the non-occurrence of the event is called the complementary event. Complementary events are events that cannot occur at the same time. When a die is thrown, the probability of getting odd and even numbers are complementary events.

7. Certain Event

An event that is sure to occur in any given experiment is a certain event. The possibility of this type of event is 1.

8. Equivalent / Identical Events

Two events are said to be equivalent or identical if any one of them implies or implied by another. The occurrence of an event implies the occurrence of another or vice versa.

Example: Even face and face 2 or face 4 or face 6 are two identical events.

8. Favorable Events

The outcomes which make necessary the happening of an event in an experiment are called favourable events.

Example: When a dice is thrown twice, the number of favourable events of getting a sum 6 is 5, i.e (3, 3), (2, 4), (4, 2), (1, 5), (5, 1)

9. Exhaustive Events

All possible outcomes of the experiments are called exhaustive events.

Example: After throwing a die there are 6 exhaustive events in the sample.

10. Independent Events and Dependent Events

If the occurrence of one event is completely unaffected by the occurrence of another event, then those are called independent events and events which are affected by others are called dependant events.

Event points, Even Space:

Let us take an experiment denoted by E. The events connected with E are called even points and the set S of all possible even points is even space of E.

Any subset A of S is obviously an event. If A contains a single point then it is a simple event, if A contains more than one point of S then A is a compound event. Then entire space S is the certain event and an empty set ∅ is an impossible event.

Solved Examples of Events in Probability

Example 1:
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?

Consider that given events are A and B
The probability of occurrence of A = P(A) = \(\frac { 4 }{ 4 + 3 } \) = \(\frac { 4 }{ 7 } \)
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(AUB) = P(A) + P(B) – P(A⋂B)
= P(A) + P(B) – P(A).P(B)
= \(\frac { 4 }{ 7 } \) + \(\frac { 3 }{ 5 } \) – \(\frac { 4 x 3}{ 7 x 5 } \)
= \(\frac { 20 + 21 – 12 }{ 35 } \) = \(\frac { 29 }{ 35 } \)

Example 2:
Two unbiased dice are rolled together. Find the odds in favor of getting 2 digits, the sum of which is 4.

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 4. Clearly, event A contains 6 equally likely events viz., (1,3), (2,2), (3,1).
Therefore, by the classical definition of probability, we get, P(A) = 3/36 = 12
Therefore, odds in favor of events A are 1:(6-1) = 1:5

FAQ’s on Events in Probability

1. What are the types of events in probability?

The different types of events are compound event, mutually exclusive event, simple event, Impossible and certain events and equally likely events.

2. What are likely events in probability?

Obtaining a number 3 on the toss of a die and getting a number 1 on the toss of a die are likely events, as the probabilities of each event are equal.

3. What is an event and give its example?

In probability, the set of outcomes from an experiment is called an event. The example is conduct an experiment by tossing a coin. The outcome of the experiment may be head or tail are events.

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