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Exhaustive Events is a set of events in which at least one of the events will take place while performing the experiment.
- Exhaustive Events represent all possible occurrences taking place in a sample space.
- An event is an outcome from a subset of sample spaces.
- Consider a sample space A for the experiment where Ei is a set of possible events for all i equal to 1,2,3 …. N.
It can be represented as E1, E2, E3, …, En are n events of a sample space A.
Ei is said to be an exhaustive event if at least one of the Eis occurs.
Mathematically, it can be written as
E1 U E2 U E3 U E4 … U En = Uni=1 Ei = A
where E1, E2, E3, …, En are known as exhaustive events.
- The most common example of exhaustive events is flipping a coin.
- While tossing a coin, there are two possible sets of outcomes, that is, head and tail.
- Such types of events are called exhaustive events.
| Table of Content |
Key Terms: Exhaustive Events, Probability, Event, Sample Space, Outcomes, Mutually Exhaustive Events, Sets
Exhaustive Events in Probability
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In probability, an event is said to be exhaustive if that event occurs at least once. Probability is defined as a branch of mathematics that deals with the occurrence of an event. It is a number between 0 and 1, where 0 represents the impossibility of an event, and 1 represents the certainty of an event.
- If the probability of an event is higher, then the event is most likely to occur.
- The concept is used in the fields of mathematics, statistics, finance and game theory.
- The formula of probability is as follows:
Probability of a required event = Number of required favourable events / total number of required events
Read More: Multiplication Theorem on Probability
Exhaustive Event Venn Diagram
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The Venn diagram for different events are as follows -
Mutually exclusive events - Two circles that do not overlap. This is because the events cannot happen at the same time.
Collectively exhaustive events - A single circle that includes all possible outcomes in the sample space.
The Venn diagram for exhaustive events for different categories is as follows:
Exhaustive Event Venn Diagram
Mutually Exclusive Events
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A mutually exclusive event is an event when both events cannot occur at the same time. It is also popular with the name disjoint events.
- It can again be explained with the help of tossing a coin example.
- When a coin is tossed, the outcome will either be head or tail, but not both.
Mutually Exclusive Events
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| Chapter-Related Articles | ||
|---|---|---|
| Experimental Probability | Elementary Event | Sum of Probabilities |
| Geometric Probability | Sure Event | Probability Impossible Events |
Mutually Exclusive and Exhaustive Events
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If Ei ∩ Ej = φ for i ≠ j i.e., events Ei and Ej are pairwise disjoint and Uni=1 Ei = A then events E1, E2, …, En are called mutually exclusive and exhaustive events.
Mutually Exclusive and Exhaustive Event
Read More: Probability distribution
Solved Examples of Exhaustive Events
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Below are some examples of exhaustive events.
Example 1: Consider a sample space E = {1, 2, 3, 4, 5, 6}. Now, let's consider three events A, B, and C associated which are as follows:
Ans:
- A: The event of getting a number greater than 3
- B: The event of getting a number greater than 2 but less than 5
- C: The event of getting a number less than 3
- These events can be written as:
- A = {4, 5, 6}, B = {3, 4}, and C = {1, 2}
- The union of all these events equals the required sample space.
- It can be written as (A ∪ B ∪ C = S).
- Hence A, B, and C are required exhaustive events.
Example 2: Consider an experiment, where a dice is rolled out and now consider the following events.
- A: event of getting a composite number
- B: event of getting an odd number
- C: event of getting 1.
Ans. In the given experiment, a dice is tossed the outcome will be:
S = { 1, 2, 3, 4, 5, 6}
- Given,
- A: event of getting composite number which is {4, 6}
- B: event of getting odd numbers which are {1, 3, 5}
- C: event of getting 1 which is {1}
- A ∪ B ∪ C is {4, 6} U {1, 3, 5} U {1}
- The result is not equal to sample space S.
- Hence the given set of events is not exhaustive.
Example 3: Two coins are tossed. Check if this experiment is exhaustive or not.
- A = the events of at least one Tail
- B = the events of at least one Head
Ans. In the given experiment, a coin is tossed twice so its possible outcomes are as follows:
- S = { HH, HT, TH, TT}
- A = the events of at least one Tail which is {HT, TH, TT}
- B = the events of at least one Head which is {HH, HT, TH}
- A U B = {HT, TH, TT} U {HH, HT, TH}
- The result is equal to sample space S.
- Hence the given set of events is exhaustive.
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Things to Remember
- An exhaustive event is an event where one of the events will definitely happen.
- Mathematically, it can be represented as A U B = S, where S is the required sample space.
- The probability of an exhaustive event is said to be 1.
- In real life, it can be explained with the example of the number of wars taking place in the world.
- In such cases, the outcome of winning and losing will definitely take place.
- The intersection of two mutually exclusive events is equal to a null value.
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Sample Questions
Ques 1. If A and B are two mutually exclusive and exhaustive events with P(B) = 4 P(A), then what is the value of P(B’)? (2 marks)
Ans. Since P(A) and P(B) are mutually exclusive and exhaustive events
- P(A) + P(B) = 1
- And P(B) = 4 P(A)
- P (A) + 4 P(A) = 1
- 5 P(A) = 1
- P(A) = ⅕
- P(B) = ⅘
- P(B’) = 1 - P(B)
- P(B’) = 1 - ⅘
- P(B’) = ⅕
Ques 2. Given an experiment of tossing a coin. Now consider an event A which includes getting a head or a tail. Check if A is an exhaustive event or not and calculate its probability. (2 marks)
Ans. When a coin is tossed the possible outcomes of the events are {Head, Tail}. It implies that whenever a coin is tossed, either a head or tail will occur. Hence, the probability of A taking will definitely occur. Hence, event A itself is the sample space. As a result, A is an exhaustive event and it will occur whenever the experiment is performed. The probability of event A is finally equivalent to one.
Ques 3. Consider the set of the first 10 natural numbers. Check if the following events are exhaustive. (3 marks)
(1) A: Selecting a prime number
(2) B: Selecting a multiple of 3
(3) C: Choosing a perfect square number
Ans. Consider a sample space which is equal to the set of the first 10 natural numbers which is S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Given that,
- A is the event of selecting a prime number as a result: A = {2, 3, 5, 7}
- B is the event of selecting a multiple of 2 as a result: B = {3, 6, 9}
- C is the event of choosing a perfect square number as a result: C = {1, 4, 9}
- Now, A ∪ B ∪ C = {2, 3, 5, 7} ∪ {3, 6, 9} ∪ {1, 4, 9}
- {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- The result is equal to sample space S.
- Hence the given set of events is exhaustive.
Ques 4. Consider an experiment, where three coins are tossed at a time, and now consider the following events where (3 marks)
(1) A: No head appears
(2) B: Exactly one head appears
(3) C: At least two heads appear
Check if the following events are exhaustive.
Ans. In the required experiment of tossing a coin three times the outcomes will be the same, i.e. the sample space constitutes the same outcomes in both cases which is as follows:
- S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
As per the given,
A: No head appears
It means only the head will appear. So, A = {TTT}
B: Exactly one head appears so B = {TTH, THT, HHT}
C: At least two heads appear
- It means, two heads and more than two heads can be considered.
C = {THH, HTH, HHT, HHH}
A ∪ B ∪ C = {TTT} ∪ {TTH, THT, HHT} ∪ {THH, HTH, HHT, HHH}
{TTT, HTT, THT, TTH, HHT, HTH, THH, HHH}
The result is equal to sample space S.
Hence the given set of events is exhaustive.
Ques 5. Consider a sample space S = {a, e, i, o, u, b, c, d, f, g}. Let A = {a, e, i, o}, B = {b, u, c, d}, C = {f, g}. Check if A, B, and C are exhaustive events. (2 marks)
Ans. As we can see A U B U C = {a, e, i, o, u, b, c, d, f, g, h, j} = S. Therefore, whenever the experiment is performed one of the events A, B, C will occur. Hence, A, B, and C are exhaustive events.
Ques 6. If A and B are two mutually exclusive and exhaustive events. If P(A) = 1/7. Find the value of 5 ×P(B). (2 marks)
Ans. Given, P(A) = 1/7
- It is given that two events are mutually exclusive and exhaustive
- So P(A ∩ B) =0 and P(A U B ) = 1
- P (A) + P(B) - P(A ∩ B) = 1
- P (A) + P(B) = 1
- P(B) = 1 - 1/7
- P(B) = 6/7
- 5 x P(B) = 30/7
Ques 7. Consider an experiment, where a dice is tossed at a time, and now consider the following events where (3 marks)
(1) A: The event of rolling a number 3
(2) B: The event of rolling a number greater than 2 but less than 4
(3) C: The event of rolling a number less than 2
Check if the following events are exhaustive.
Ans. When a dice is tossed the sample obtained is S = { 1, 2, 3, 4, 5, 6}
- Given,
- A: The event of rolling a number greater 3 which is {3}
- B: The event of rolling a number greater than 2 but less than 4 which is {3}
- C: The event of rolling a number less than 2 which is {1}
- A U B U C = {1, 3}
- This is not equivalent to sample space S
- Hence the following events are not exhaustive.
Ques 8. Consider an experiment, where two coins are tossed at a time, and now consider the following events where (2 marks)
(1) A: No tail appears
(2) B: Exactly one tail appears
(3) C: At least two tails appear
Check if the following events are exhaustive.
Ans. In the required experiment of tossing a coin two times the outcomes will be the same, i.e. the sample space constitutes the same outcomes in both cases which is as follows:
- S = {HH, HT, TH, TT}
As per the given,
A: No head appears
It means only tails will appear. So, A = {TT}
B: Exactly one tail appears so B = {TH, HT}
C: At least two tails appear
- It means, two tails and more than two tails can be considered.
C = {TH, HT, TT}
A ∪ B ∪ C = {TT} ∪ {TH, HT} ∪ {TH, HT, TT}
{HH, HT, TH, TT}
The result is equal to sample space S.
Hence the given set of events is exhaustive.
Ques 9. Consider an experiment, where a dice is rolled out and now consider the following events. (3 marks)
(1) A: event of getting a prime number
(2) B: event of getting an even number
(3) C: event of getting 1.
Ans. In the given experiment, a dice is tossed the outcome will be:
S = {1, 2, 3, 4, 5, 6}
- Given,
- A: event of getting prime number which is { 2, 3, 5}
- B: event of getting even numbers which are {2, 4, 6}
- C: event of getting 1 which is {1}
- A ∪ B ∪ C is { 2, 3, 5} U {2, 4, 6} U {1}
- The result is equal to sample space S.
- Hence the given set of events is exhaustive.
Ques 10. Consider an experiment, where a coin is tossed at a time, and now consider the following events where (3 marks)
(1) A: No tail appears
(2) B: No head will appear
(3) C: Both head and tail will appear
Check if the following events are exhaustive.
Ans. In the required experiment of tossing a coin two outcomes will be achieved, i.e. the sample space constitutes the same outcomes in both cases which is as follows:
- S = {H, T}
As per the given,
A: No tail appears
It means only the head will appear. So, A = {H}
B: No head will appear so B = {T}
C: Both head and tail will appear
- C = {H, T}
A ∪ B ∪ C = {H} ∪ {T} ∪ {H, T}
{H, T}
The result is equal to sample space S.
Hence the given set of events are exhaustive.
Ques 11. Consider an experiment, where a dice is rolled out and now consider the following events. (2 marks)
(1) A: event of getting an odd number
(2) B: event of getting an even number
Ans. In the given experiment, a dice is tossed the outcome will be:
S = { 1, 2, 3, 4, 5, 6}
- Given,
- A: event of getting an odd number which is { 1, 3, 5}
- B: event of getting even numbers which is {2, 4, 6}
- A ∪ B is { 1, 3, 5} U {2, 4, 6}
- The result is equal to sample space S.
- Hence the given set of events are exhaustive.
Ques 12. If A, B and C are three mutually exclusive and exhaustive events with P(A) = 3 P(B) = 4 P(C), then P(B∪C)=. (2 marks)
Ans. P(A) +P(B)+P(C)= 1
- P(A) + P(A)/3 + P(A)/4 = 1
- 12 P(A) + 4 P(A) + 3 P(A) = 12
- 19 P(A) = 12
- P(A) = 12 / 19
- P(B) = 4 / 19
- P(C) = 3 /19
- P(B U C) = P(B) + P(C)
- 4 / 19 + 3 / 19
- 7 / 19
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