Science > Mathematics > Statistics and Probability > Probability > Events and Their Types
In the last article, we have studied the concept of the sample space. In this article, we shall understand the concept of events in detail.
Concept of Event:
Any subset of a sample space is called an event. The event is denoted by a capital letter.
Example – 1:
A = an event, that a card selected from a pack of cards is a Diamond.
A Diamond card can be obtained in 13C1 ways =13 ways
Number of cases favourable to the event A = 13
∴ n(A) = 13
Example – 2:
B = an event, that 2 cards selected are consisting of one King and other a Queen.
A King can be obtained in 4C1 ways = 4 ways.
After the selection of a King in any ‘one of four ways,
the selection of a queen can be made in 4C1 ways = 4 ways.
∴ n(B) = 4 x 4 = 16
Example – 3:
C = an event, that a die shows a number greater than 3
∴ C = { 4, 5, 6}
∴ n(C) = 3
The Terminology of Events:
The Occurrence of an Event:
An event associated with a random experiment is said to occur if any one of the elementary events associated with it is an outcome.
Thus, if an elementary event E is an outcome of a random experiment and A is an event such that E ∈ A, then we say that the event A has occurred.
The Negation of an Event:
Corresponding to every event A associated to a random experiment, we define event ‘not A’ which is said to occur when and only when A does not occur.
If A is an event that even number comes up when a die is thrown; then ‘not A’ i.e. A’ is an event that even number does not come up.
Impossible Event :
If an event is an empty set, then the event is called an impossible event and is denoted by Φ.
Example – 1:
A = an event that the score on two dice is 15.
Example – 2:
B= an event of having a quadratic equation of three roots.
Certain or Sure Event :
If an event is the same as the sample space of the experiment then the event is called a certain event or sure event.
Example – 1:
A = an event that a card selected from a pack of 52 cards is either a red card or a black card.
In this case n(A) = 52 = n(S). and A = S
Example – 2:
B = an event that a die shows a number which is odd or even.
B = {1, 2,3,4,5,6}
∴ n(B) = 6 = n(S) and B = S
Elementary event:
If an event contains only one sample point, then the event is called an elementary event or a simple event.
Example – 1:
A = an event, card selected is a queen of hearts.
A queen of hearts can be obtained in one way.
∴ n(A) = 1.
∴ A is an elementary event.
Example – 2:
B = an event that the score on two dice is 12.
B = {(6,6)}
∴ n(B) = 1
∴ B is an elementary event.
The Complement of an event A:
Let A be an event of a sample space S then the event consisting of all the cases of the samples space which are not favourable to the event is called the complement of the event A and is denoted by A’ or
∴ A’ = {x | x ∈ S, but x ∉ A}
Examples:
- A = an event that card selected is a spade. ∴ A’ = an event that card selected is not spade
- B = an event that the score on the two dice is greater than 4. ∴ B’ = an event that the score on the two dice is less than or equal to 4.
- C = an event that, the room is lighted. ∴ C’ = an event that, the room is not lighted.
- D = an event that, India wins at least one game. ∴ D’ = an event that, India does not win any game.
- E = an event, that the room has at least one fan. ∴ E’ = an event, that the room has no fan.
Notes:
- If A = an event consisting of at least one then A’ = an event consisting of none.
- If S contains n sample points and A contains m sample points. Then A’ will contain (n – m) sample points.
Important Results:
| (a) Φ’ = S | (b) S’ = Φ |
| (c) A U A’ = S | (d) A ∩ A’ = Φ |
| (e) n(A) + n(A’) = n(S) |
Compound Event:
A compound event is one in which there is more than one possible outcome. Determining the probability of a compound event involves finding the sum of the probabilities of the individual events and, if necessary, removing any overlapping probabilities.
Mutually Exclusive Events:
Two or more events associated with a random experiment are said to be mutually exclusive or incompatible events if the occurrence of any of them prevents the occurrence of all others. i.e. no two or more of them can occur simultaneously in the same trial.
Example – 1:
Consider an experiment of rolling a die. Sample space for the experiment is
S= {1, 2, 3, 4, 5, 6}
Let A = Event of getting even number
∴ A = {2, 4, 6}
B = Event of getting odd number
∴ B = {1, 3, 5}
C = Event of getting multiple of 3
∴ C = {3, 6}
Now A ∩ B = Φ , Hence, A and B are mutually exclusive events.
Now B ∩ C ≠ Φ, Hence, B and C are not mutually exclusive events.
Now A ∩ C ≠ Φ, Hence, A and C are not mutually exclusive events.
Now A ∩ B ∩ C = Φ, Hence, A, B and C all taken together are mutually exclusive events.
Example – 2:
Consider an experiment of drawing two cards from a well-shuffled pack of 52 cards.
Let A = Event of getting both red cards and B = Event of getting both black cards
As two cards drawn can not be red and black simultaneously.
Hence these two events are mutually exclusive events.
Notes:
- The probability of mutually exclusive events is a sum of their individual probabilities. i.e. If A ∩ B = Φ, i.e. A and B are mutually exclusive events then P(A U B) = P(A) + P(B).
- To show the two events are mutually exclusive, show that A ∩ B = Φ i.e. P(A or B) = P(A U B)= P(A) + P(B).
Exhaustive Events:
Two or more events associated with a random experiment are exhaustive if their union is a sample space.
Example – 1:
Consider an experiment of rolling a die. Sample space for the experiment is
S= {1, 2, 3, 4, 5, 6}
Let A = Event of getting even number
A = {2, 4, 6}
B = Event of getting odd number
B = {1, 3, 5}
A U B = {1, 2, 3, 4, 5, 6} = S
Hence A nad B are exhaustive events.
Example – 2 :
Consider an experiment of drawing two cards from well shufflled pack of 52 cards.
Let A = Event of getting both red cards
B = Event of getting both black cards
A U B = S
Hence A nad B are exhaustive events.
- Probability of sum of exhaustive events is always 1. i.e. If A, B, C , …. are exhaustive events then P(A) + P(B) + P(C) + ……= 1
- To show the two events are exhaustive events, show that sum of their probability is 1.
- Two events may be exhaustive and mutually exclusive.
Example: exhaustive and mutually exclusive events.
Consider an experiment of rolling a die. Sample space for the experiment is
S= {1, 2, 3, 4, 5, 6}
Let, A = Event of getting even number
A = {2, 4, 6}
B = Event of getting odd number
B = {1, 3, 5}
A U B = {1, 2, 3, 4, 5, 6} = S and A ∩ B = Φ
Hence A nad B are exhaustive and mutually exclusive events.
Example: exhaustive and non mutually exclusive events.
Consider an experiment of rolling a die. Sample space for the experiment is
S= {1, 2, 3, 4, 5, 6}
Let, A = Event of getting even number
A = {2, 4, 6}
B = Event of getting odd number or perfect square
B = {1, 3, 5, 4}
A U B = {1, 2, 3, 4, 5, 6} = S and A B ≠ Φ
Hence A nad B are exhaustive and non mutually exclusive events.
Symbolic Meaning of Compound Events
Here we have to understand how to create a new event using two or more given events associated with the random experiment
| Verbal description of the event | Symbolic representation |
| A, B, C | The main events |
| Not A | A’ |
| A or B | A ∪ B |
| At least one of A or B | A ∪ B |
| A and B | A ∩ B |
| A but not B | A ∩ B’ |
| Neither A nor B | A’ ∩ B’ or (A U B)’ |
| At least one of A, B, or C | A U B U C |
| All the three A, B and C | A ∩ B ∩ C |
| A but not B and not C | A ∩ B’ ∩ C’ |
| Exactly one of A, B and C | (A∩ B’ ∩ C’) ∪(A’ ∩ B ∩ C’) ∪(A’ ∩B’ ∩ C) |
| Exactly two of A, B and C | (A∩B ∩ C’) ∪(A ∩B’ ∩ C)∪(A’ ∩ B ∩ C) |
| None of A, B and C | A’ ∩ B’ ∩ C’ or (A U B U C)’ |
| At least one of A, B, or C | 1 – (A ∩ B ∩ C ) |
| Not more than two occurs | (A ∩ B)U(B ∩ C)U(C ∩ A) – 3(A∩B∩C) |
In the next article, we shall study some basic problems of probability based on tossing of coins.