Exhaustive Events in Probability | Mutually Exclusive and Exhaustive Events | Solved Examples, FAQs

Class 11 Mathematics | Written by Neeraj Anand
Published by ANAND TECHNICAL PUBLISHERS

In probability theory, understanding different types of events is essential to solve problems accurately. One of the most important concepts is that of Exhaustive Events. This concept plays a crucial role in understanding how probabilities cover all possible outcomes of an experiment.

What are Exhaustive Events in Probability?

Exhaustive events in probability refer to a collection of events that together cover all possible outcomes of an experiment or situation.

Mathematically, if E₁​, E₂​, ……., E_n​ are exhaustive events, their union (E₁​ ∪ E₂​ ∪ …… ∪E_n​) equals the entire sample space (S).

E₁​ ∪ E₂​ ∪ …… ∪ E_n = S

In other words, events E₁, E₂, E₃, …,E_n are called exhaustive events if at least one of them necessarily occurs whenever the experiment is performed.

What are Exhaustive Events in Probability?

Let us consider the experiment of throwing a die.

Sample space S = {1, 2, 3, 4, 5, 6}

Assume that A, B and C are the events associated with this experiment. Also, let us define these events as:

A be the event of getting a number greater than 3

B be the event of getting a number greater than 2 but less than 5

C be the event of getting a number less than 3

We can write these events as:

A = {4, 5, 6}

B = {3, 4}

and C = {1, 2}

We observe that

A ⋃ B ⋃ C = {4, 5, 6} ⋃ {3, 4} ⋃ {1, 2} = {1, 2, 3, 4, 5, 6} = S

Therefore, A, B, and C are called exhaustive events.

However, the probability of exhaustive events is equal to 1.

Collectively Exhaustive Events

Collectively exhaustive events, in probability theory, refer to a set of events that cover all possible outcomes and no outcome is counted more than once across the events. Here, the events do not overlap with each other.

Consider flipping a fair coin. The possible outcomes are either heads (H) or tails (T). In this case, the events “getting heads” and “getting tails” are collectively exhaustive because one of these outcomes must happen when the coin is flipped. There are no other possible outcomes besides heads or tails.

Mutually Exclusive and Exhaustive Events

Mutually exclusive events and exhaustive events are two important concepts in probability theory.

Aspect	Mutually Exclusive Events	Exhaustive Events
Definition	Events that cannot occur simultaneously.	Events that cover all possible outcomes of an experiment.
Occurrence	Cannot occur together in the same trial. A1 ⋂ A2 ⋂ … ⋂ An= φ	At least one of the events must occur A1 ⋃ A2 ⋃ … ⋃ An= S
Probability Relationship	The intersection of mutually exclusive events has zero probability i.e., P(A ⋂ B) = 0.	The union of exhaustive events covers the entire sample space i.e., P(A1 ⋃ A2 ⋃ … ⋃ An)= 1.
Example	Rolling an even number and rolling an odd number on a fair six-sided die.	Rolling a number less than 3 and rolling a number greater than or equal to 3 on a fair six-sided die.

Example: Consider rolling a fair six-sided die. The possible outcomes are numbers 1 through 6. Now, let’s define two events:

• Event A: Rolling an odd number {1, 3, 5}
• Event B: Rolling an even number {2, 4, 6}

Solution:

• Events A and B are mutually exclusive because if you roll an odd number (Event A), you cannot roll an even number (Event B), and vice versa.
• Events A and B are also collectively exhaustive because together, they cover all possible outcomes of rolling the die. You will always roll either an odd number (Event A) or an even number (Event B).

Exhaustive Event Venn Diagram

Exhaustive Events Venn Diagram

The below figure shows the Venn diagram representation of collectively exhaustive events in comparison with exclusive events.

Examples of Exhaustive Events

Examples of Exhaustive Events can be coin tossing, rolling a dice and drawing cards from a deck of cards. Explanation of each of them is given below:

Coin Tossing

• When tossing a fair coin, the possible outcomes are either heads (H) or tails (T).
• The events “getting heads” and “getting tails” are collectively exhaustive because one of these outcomes must occur.
• For example, if you toss a coin, you will either get heads or tails. There are no other possible outcomes.

Rolling a Dice

• When rolling a fair six-sided die, the possible outcomes are numbers 1 through 6.
• The events “rolling a 1,” “rolling a 2,” and so on up to “rolling a 6” are collectively exhaustive because one of these outcomes must occur.
• For instance, if you roll a die, you will get a number between 1 and 6. There are no other possible outcomes besides these six numbers.

Drawing Cards from a Deck

• When drawing cards from a standard deck of 52 playing cards, the possible outcomes include the various ranks (2 through 10, Jack, Queen, King, Ace) and suits (hearts, diamonds, clubs, spades).
• The events “drawing a heart,” “drawing a diamond,” “drawing a club,” and “drawing a spade” are collectively exhaustive because one of these outcomes must occur.
• Additionally, the events “drawing a 2,” “drawing a 3,” and so on up to “drawing an Ace” for each suit are also collectively exhaustive.
• For example, if you draw a card from a deck, it will be either a heart, diamond, club, or spade, and it will also be one of the ranks from 2 to Ace. There are no other possible outcomes besides these.

Exhaustive Events Venn Diagram

The below figure shows the Venn diagram representation of collectively exhaustive events in comparison with exclusive events.

Solved Examples on Exhaustive Events in Probability

Example 1:

In an experiment, three coins are tossed at a time, consider the following events.

A: ‘No tail appears’,

B: ‘Exactly one tail appears’

C: ‘At least two tails appear’

Do they form a set of exhaustive events?

Solution:

In the experiment of tossing a coin three times or three coins tossed at a time, the outcomes will be the same, i.e. the sample space constitutes the same outcomes in both the cases.

Thus,

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

As per the given,

Event A: ‘No tail appears’

That means only heads will appear.

So, A = {HHH}

Event B: ‘Exactly one tail appears’

So, B = {HHT, HTH, THH}

Event C: ‘At least two tails appear’

That means, two tails and more than two tails can be considered.

So, C = {HTT, THT, TTH, TTT}

Now,

A ∪ B ∪ C = {HHH} ∪ {HHT, HTH, THH} ∪ {HTT, THT, TTH, TTT}

= {TTT, HTT, THT, TTH, HHT, HTH, THH, HHH}

= S

Therefore, A, B and C are exhaustive events.

Example 2:

Consider the set of first 10 natural numbers. Check if the following defined events are exhaustive.

A: Selecting a prime number

B: Selecting a multiple of 2

C: Choosing a perfect square number

Solution:

Sample space = Set of first 10 natural numbers

i.e. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Given that,

A is the event of selecting a prime number.

So, A = {2, 3, 5, 7}

B is the event of selecting a multiple of 2.

So, B = {2, 4, 6, 8, 10}

C is the event of choosing a perfect square number.

So, C = {1, 4, 9}

Now,

A ∪ B ∪ C = {2, 3, 5, 7} ∪ {2, 4, 6, 8, 10} ∪ {1, 4, 9}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

= S

Hence, the given set of events are exhaustive.

Example 3:

You roll two fair six-sided dice. Define events for the sum of the numbers rolled (2 through 12). Are these events collectively exhaustive? What is the probability of getting any sum from 6 to 12?

Solution:

Sum of the numbers rolled can range from 2 (when both dice show a 1) to 12 (when both dice show a 6).

Here are the events for each possible sum:

• Event for the sum of 2: {(1, 1)}
• Event for the sum of 3: {(1, 2), (2, 1)}
• Event for the sum of 4: {(1, 3), (2, 2), (3, 1)}
• Event for the sum of 5: {(1, 4), (2, 3), (3, 2), (4, 1)}
• Event for the sum of 6: {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)}
• Event for the sum of 7: {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}
• Event for the sum of 8: {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}
• Event for the sum of 9: {(3, 6), (4, 5), (5, 4), (6, 3)}
• Event for the sum of 10: {(4, 6), (5, 5), (6, 4)}
• Event for the sum of 11: {(5, 6), (6, 5)}
• Event for the sum of 12: {(6, 6)}

These events cover all possible sums from 2 to 12. Therefore, they are collectively exhaustive.

To find the probability of getting any sum from 6 to 12, we need to calculate the probability of each event and then add them up.

Since each die is fair and has 6 sides, there are a total of (6 × 6 = 36) equally likely outcomes when rolling two dice.

Calculate the probabilities for each event:

• Probability of sum = 6: P(6) = 5/36​
• Probability of sum = 7: P(7) = 6/36​
• Probability of sum = 8: P(8) = 5/36​
• Probability of sum = 9: P(9) = 4/36​
• Probability of sum = 10: P(10) = 3/36​
• Probability of sum = 11: P(11) = 2/36
• Probability of sum = 12: P(12) = 1/36​

Now, to find the probability of getting any sum from 6 to 12, we add up the probabilities of all these events:

P(sum from 6 to 12)=P(6)+P(7)+P(8)+…+P(12)

⇒ P(sum from 6 to 12)=536+636+536+436+336+236+136365​+366​+365​+364​+363​+362​+361​

⇒ P(sum from 6 to 12)=5+6+5+4+3+2+136365+6+5+4+3+2+1​

⇒ P(sum from 6 to 12)=26​/36

⇒ P(sum from 6 to 12)=13/18

So, the probability of getting any sum from 6 to 12 when rolling two fair six-sided dice is 13/18 .

Example 4:

You flip three fair coins. Define events for the number of heads obtained (0, 1, 2, or 3). Are these events collectively exhaustive? What is the probability of getting any number of heads from 0 to 3?

Solution:

To define events for the number of heads obtained when flipping three fair coins, we need to consider all possible combinations of outcomes. Each coin can either show heads (H) or tails (T).

Here are the events for each possible number of heads:

• Event for 0 heads (all tails): {TTT}
• Event for 1 head: {HTT, THT, TTH}
• Event for 2 heads: {HHT, HTH, THH}
• Event for 3 heads (all heads): {HHH}

These events cover all possible outcomes when flipping three fair coins and represent the number of heads obtained (0, 1, 2, or 3). Therefore, they are collectively exhaustive.

Calculate the probability of getting any number of heads from 0 to 3.

Since each coin is fair and has 2 equally likely outcomes (heads or tails), there are a total of 23=8 equally likely outcomes when flipping three coins.

• Probability of getting 0 heads: There is only 1 outcome (TTT) with 0 heads. P(0 heads)= 1/8
• Probability of getting 1 head: There are 3 outcomes (HTT, THT, TTH) with 1 head. P(1 head)= 3/8
• Probability of getting 2 heads: There are 3 outcomes (HHT, HTH, THH) with 2 heads. P(2 heads)=3/8
• Probability of getting 3 heads: There is only 1 outcome (HHH) with 3 heads. P(3 heads)= 1/8

To find the probability of getting any number of heads from 0 to 3, we add up the probabilities of these events:

(0 to 3 heads)=P(0 heads)+P(1 head)+P(2 heads)+P(3 heads)(0 to 3 heads)=P(0 heads)+P(1 head)+P(2 heads)+P(3 heads)

⇒P(0 to 3 heads)=18+38+38+18P(0 to 3 heads)=81​+83​+83​+81​

⇒P(0 to 3 heads)=88=1P(0 to 3 heads)=88​=1

So, the probability of getting any number of heads from 0 to 3 when flipping three fair coins is 1 or 100%. This is because one of these outcomes is guaranteed to occur when you flip the coins.

Practice Questions on Exhaustive Events

Question 1: You flip a fair coin. Define two events:

• Event A: Getting heads (H)
• Event B: Getting tails (T)

Are these events collectively exhaustive? What is the probability of getting either heads or tails?

Question 2: You roll a fair six-sided die. Define six events, one for each possible outcome (numbers 1 through 6). Are these events collectively exhaustive? What is the probability of rolling any number from 1 to 6?

Question 3: You draw a card from a standard deck of 52 playing cards. Define four events:

• Event A: Drawing a heart
• Event B: Drawing a diamond
• Event C: Drawing a club
• Event D: Drawing a spade

Are these events collectively exhaustive? What is the probability of drawing a card of any suit?

Question 4: You flip two fair coins. Define three events:

• Event A: Getting two heads (HH)
• Event B: Getting two tails (TT)
• Event C: Getting one head and one tail (HT or TH)

Are these events collectively exhaustive? What is the probability of getting either two heads, two tails, or one head and one tail?

Exhaustive Events: Frequently Asked Questions

What are exhaustive events in probability?

Exhaustive events in probability are a set of events that cover all possible outcomes of an experiment. This means that at least one of the events must occur, and together they complete the entire sample space.

What are mutually exhaustive events?

Mutually exhaustive events are events that cannot occur simultaneously, but together they cover all possible outcomes i.e. if the union of mutually exclusive events forms a complete sample space it is said to be mutually exhaustive event.

What are collectively exhaustive events?

Collectively exhaustive events are events that together cover all possible outcomes of an experiment. This means that at least one of the events must occur, and they encompass the entire sample space.

What is difference between mutually exclusive and exhaustive events?

• Mutually exclusive events cannot occur simultaneously, while exhaustive events ensure that at least one event must occur.
• Mutually exclusive events cover separate and non-overlapping portions of the sample space, while exhaustive events collectively cover the entire sample space.

What is difference between exhaustive events and sample space?

• Exhaustive events are a set of events that together cover all possible outcomes of an experiment, ensuring that one of them must occur.
• The sample space, on the other hand, is the set of all possible outcomes of an experiment, including both the outcomes covered by the exhaustive events and any other outcomes not explicitly considered as events.

Are all exhaustive events mutually exclusive?

Exhaustive events can be mutually exclusive if they cover separate and non-overlapping portions of the sample space. However, they can also overlap or intersect, depending on the specific experiment or situation. It’s possible for exhaustive events to be mutually exclusive, but it’s not a requirement.

What is non exhaustive event?

A non-exhaustive event is an event that does not cover all possible outcomes of an experiment. In other words, there are additional outcomes beyond those encompassed by the event.

Can events be exhaustive and independent?

Yes, events can be exhaustive (covering all possible outcomes) and independent (occurring without influencing each other) simultaneously within a given probability space..

What is the meaning of exhaustive events?

A set of events are called exhaustive events if at least one of them necessarily occurs whenever the experiment is performed. Also, the union of all these events constitutes the sample space of that experiment.

What does it mean when two events are exhaustive?

Two events are exhaustive when their union is equal to the sample space.

What is the difference between mutually exclusive and exhaustive events? Explain with an example.

If A ∩ B = φ for i.e., events A and B are disjoint and A ∪ B = S, then events A and B are called mutually exclusive and exhaustive events. For example, in an experiment of rolling a die, the events denoting the occurrence of even and odd numbers are disjoint yet they cover all the outcomes of the sample space when we take union of these events.

Are all exhaustive events mutually exclusive?

No, all exhaustive events may or may not be mutually exclusive.

Can events be exhaustive and independent?

Yes, events are exhaustive and independent. Suppose two events A and B are exhaustive, i.e. A ∪ B = S are also can be independent if and only if P(A ∩ B) = P(A) P(B).

Summary

🔍 What Are Exhaustive Events?

A set of events is said to be exhaustive if it includes all possible outcomes of an experiment. In simple terms, when every possible result of an experiment is covered by a set of events, those events are considered exhaustive.

• Mathematically, if an experiment has a sample space SSS, and the events E1,E2,E3,…,EnE_1, E_2, E_3, \dots, E_nE1​,E2​,E3​,…,En​ cover all possible outcomes of SSS, then the events are exhaustive.
• This means that at least one of the events must occur when the experiment is performed.

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✅ Examples of Exhaustive Events

1. Rolling a Die

• The possible outcomes are: {1,2,3,4,5,6}\{1, 2, 3, 4, 5, 6\}{1,2,3,4,5,6}
• Events like getting a number less than or equal to 6 form an exhaustive set since they cover all possible outcomes.

2. Tossing a Coin

• The possible outcomes are: {Heads,Tails}\{Heads, Tails\}{Heads,Tails}
• The events “getting a head” or “getting a tail” are exhaustive because together, they cover all possible outcomes of the toss.

3. Drawing a Card from a Deck

• The possible outcomes are all 52 cards in the deck.
• Events like “drawing a red card” or “drawing a black card” are exhaustive since they cover all possible cards in the deck.

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🔑 Key Characteristics of Exhaustive Events

• The sum of the probabilities of exhaustive events is always equal to 1.
• Exhaustive events cover the entire sample space of the experiment.
• They ensure that at least one event will always occur when the experiment is performed.

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📌 Importance in Probability

Understanding exhaustive events helps in solving problems related to the total probability theorem and aids in determining the likelihood of various outcomes occurring. It is particularly important for:

• Board exams (CBSE Class 11 Mathematics)
• Competitive exams like JEE Mains & Advanced

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