Events in Probability

Events in Probability- In Probability, an event can be defined as any outcome or set of outcomes from a random experiment. In other words, an event in probability is a subset of the respective sample space.

Example:
1. If you roll a die, the event could be "getting a 3" or "getting an even number."
2. If you toss two coins simultaneously , the event could be getting "getting at least 1 heads" or "getting two tails".

This concept of events is fundamental to understanding probability theory

In this article, we will learn about events in Probability, including their types, definitions, classification, and the algebra of events, etc.

Sample Space

A Sample Space is the set of all possible outcomes of an experiment or a random phenomenon. Sample Space is denoted by the symbol "S" and represents all the possible outcomes that can occur.

Example: When flipping a coin, the sample space is {heads, tails}, because those are the only two possible outcomes. Similarly, when rolling a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}, because those are the only possible outcomes.

Types of Events in Probability

It is essential to understand the various types of events that can occur during the execution of random experiments. There are various types of events in probability, which are discussed as follows:

• Impossible and Sure Events
• Simple Event and Compound Event
• Dependent and Independent Events
• Mutually Exclusive Events
• Exhaustive Events
• Equally Likely Events

➣ Impossible Event and Sure Event: An event which cannot happen under any circumstances is called an impossible event. Its probability is 0, whereas a sure event (also called a certain event) is an event that is guaranteed to happen, and its probability is always 1.

Examples of Impossible Events:

• Rolling a 7 on a standard 6-sided die.
• Having 30th February in a year.

Examples of Sure Events:

• Rolling a number less than 7 on a die.
• The sun rising in the east.

➣ Independent Event and Dependent Event: Independent events are those in which the probability of an event remains the same, regardless of previous outcomes. Whereas, dependent events are those in which the probability of an event changes based on previous outcomes

Examples of Dependent Events:

Example 1: Drawing two cards from a deck without replacement.

If you draw one card and do not replace it, the total number of cards in the deck changes. The probability of drawing a specific card on the second draw is affected by the outcome of the first draw, hence they are dependent events.

Example 2: Picking a marble from a bag, not replacing it, and then picking another marble.

• If the first marble is not replaced, the total number of marbles changes, which influences the probability of picking the second marble. Hence, the events are dependent.

Examples of Independent Events:

Example 1: Flipping a coin twice.

The outcome of the first flip (heads or tails) does not affect the outcome of the second flip. The probability of each flip remains 1/2, so the events are independent.

Example 2: Rolling a die and flipping a coin.

The result of rolling the die (e.g., getting a 4) has no impact on the result of flipping the coin (heads or tails). Both events are independent.

Learn more about: [Dependent and Independent Events]

➣ Simple and Compound Events: When an event consists of only one point of the sample space, this event is called a simple event, and events with two or more points of the sample space are called compound events.

Examples of Simple Events:

Example 1: Tossing a Coin

Sample space: {Heads, Tails}.
Getting "Heads" is a simple event.

Example 1: Rolling a Die

Sample space: {1, 2, 3, 4, 5, 6}.
Getting a 4 is a simple event.

Example of Compound Event:

Scenario: Rolling a fair six-sided die

Let the sample space S be all possible outcomes when rolling the die:
S = {1, 2, 3, 4, 5, 6}

Event A: Rolling an even number (i.e., 2, 4, or 6). So, we can represent it as a set:
A = {2, 4, 6}

Event B: Rolling a number greater than 3 (i.e., 4, 5, or 6). So, we can represent it as a set:
B = {4, 5, 6}

Compound Event (Union of A and B): {2,4,5,6}

➣ Mutually Exclusive Events:  Mutually exclusive events have no outcomes in common. In other words, if one event happens, the other cannot happen.

Example:

Let's consider a scenario of flipping a fair coin.

Event A: The coin lands on heads .A = {Heads}
Event B: The coin lands on tails. B = {Tails}

The intersection of mutually exclusive events is the empty set: A∩B = ∅.

➣ Exhaustive Events:  The collection of those events is exhaustive, covering all the possible outcomes.

Example:

Consider a random experiment where a coin is tossed twice. The sample space (S) is:
S = {HH, HT, TH, TT}

Define the following events:

A: Getting at least one head A = {HH, HT, TH}
B: Getting two tails B = {TT}

The union of A and B gives: A ∪ B = {HH, HT, TH, TT} = S
This means the events A and B together cover all possible outcomes in the sample space S. Hence, A and B are exhaustive events.

➣ Equally Likely Events: Equally likely events have the same probability of occurring. In a random experiment, all outcomes are equally likely if none is favored over the others.

Example:

In the case of rolling a fair six-sided die, there are six equally likely outcomes: 1, 2, 3, 4, 5, and 6. Since all outcomes are equally likely, the probability of rolling any specific number is 1/6.

Learn more: [Types of Events in Probability]

Algebra of Events

Since events are nothing but subsets of sample space, two or more events can be combined using four different operations, union, intersection, difference, and complement.. Let's consider three events A, B, and C defined over the sample space S. 

Union of Events in Probability

Let's say we have two events, A and B, then the union of A and B represents the event that occurs if either A or B (or both) occurs. Union of Events A and B is denoted by A ∪ B, and it contains all the outcomes that are in either A, B, or both.

Let's consider an example for an Intersection of Events, let two events be E₁ = {2, 3, 4, 5} and E₂ = {3, 4, 7, 8}.

The union of E₁ and E₂​ is denoted by E₁∪E₂​, and it includes all the elements from both sets without duplication.

Thus, E₁​ ∪ E₂​ = {2, 3, 4, 5, 7, 8}

Intersection of Events in Probability

Let's say we have two events, A and B, then the intersection of A and B represents the event that occurs if both A and B occur simultaneously. The intersection of Events A and B is denoted by A ∩ B.

Let's consider an example for an Intersection of Events, let two events be E₁ = {2, 3, 4, 5} and E₂ = {3, 4, 7, 8}. Also, assume that the intersection of both events is represented by E, i.e., E = E₁ ∩ E₂, that:

Thus, E = E₁ ∩ E₂ = {3, 4}

Complimentary Event 

For every event A, there exists another event A', which is called a complementary event.  It consists of all those elements that do not belong to event A. For example, In the rolling die experiment. Let's say event A is defined as getting an even number. So, A = {2,4,6} 
The complement A' of event A will consist of all the elements in the sample space that are not in event A. Thus, A' = { 1,3,5}

Difference of Events in Probability

For any two events A and B, the difference of A and B represents the event that consists of all the outcomes which are in A but not in B. The Difference of events A and B is denoted as A - B.

Event A or B

The Union of two sets A and B is denoted as A ∪ B. This contains all the elements that are in either set A, set B, or both. This event A or B is defined as, 

Event A or B = A ∪ B
A ∪ B = {w : w ∈ A or w ∈ B}

Events A and B

The intersection of two sets A and B is denoted as A ∩ B. This contains all the elements that are in both set A and set B. Events A and B are defined as, 

Event A and B = A ∩ B
A ∩ B= {w: w ∈ A and w ∈ B}

Event A but not B

The set difference A - B consists of all the elements that are in A but not in B. The events A but not B are defined as, 

A but not B = A - B 
A - B = A ∩ B'

Where B' is the complement of event B.

How to Find the Probability of an Event

We can easily find the probability of an event by following the steps discussed below,

• Step 1: Find the total sample space of the experiment.
• Step 2: Find the number of favourable outcomes of the experiment.
• Step 3: Use the formula to calculate the probability as,

Probability = (Favourable Outcome)/(Total Outcome)

Read More,

• Probability Distribution
• Chance and Probability
• Union of Sets

Sample Problems on Events in Probability

Here we have provided you with a few solved sample problems on events in probability:

Problem 1: A die is thrown in the game of Ludo, and E₁ denotes the event of getting even numbers and E₂ represents the event of getting a number more than 3. Find the Set for the following events,

1. E₁ or E₂
2. E₁ and E₂

Solution:

The sample space for the die will be,
S= {1, 2, 3, 4, 5, 6}

• E₁ (only even numbers) = {2, 4, 6}
• E₂ (number more than 3) = {4, 5, 6}

1. E₁ or E₂ = {2, 4, 5, 6}
2. E₁ and E₂ = {4, 6}

Problem 2: A die is thrown and the set for the sample space obtained is, S = {1, 2, 3, 4, 5, 6} E₁ is defined as the event of obtaining a number less than 5 and E₂ is defined as the event of obtaining a number more than 2.
Find the set for the following,

1. E₁ but not E₂
2. E₂ but not E₁

Solution:

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

• E₁ (a number less than 5)= {1, 2, 3, 4}
• E₂ (a number more than 2)= {3, 4, 5, 6}

1. E₁ but not E₂ = {1, 2}
2. E₂ but not E₁ = {5, 6}

Problem 3: Write the sample space for tossing three coins at once, also answer the event of 2 exactly 2 heads at a time.

Solution:

Tossing Three Coins the sample space is,
S = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

Hence, the Sample Space Comprises 6 Possible Outcomes
Event (E) for the occurrence of exactly two heads,

E = {(H, H, T), (H, T, H), (T, H, H)}

Problem 4: Name the types of events obtained from the experiments given below.

1. A coin is tossed for the 5th time,, and in the event of getting a tail when the first four times, the result was ahead.
2. S (sample space)= {1, 2, 3, 4, 5} and E= {4}
3. S= {1, 2, 3, 4, 5} and E= {2, 4}
4. S= {1, 2, 3, 4, 5}, E₁= {1, 2} and E₂= {3, 4}

Solution:

1. No matter how many times the coin is tossed, every time the probability of getting a tail will be 0.5 irrespective of the previous outcomes, therefore the event will be an independent event.

2. E= {4} is a Simple event.

3. E= {2, 4} is a compound event.

4. E₁ and E₂ are Mutually exclusive events.

Problem 5: The sample Space of an experiment is given as,

S = {10, 11, 12, 13, 14, 15, 16, 17} and the event, E is defined as all the even numbers. What will be the complementary event for E?

Solution:

S = {10, 11, 12, 13, 14, 15, 16, 17}
E (All even numbers) = {10, 12, 14, 16}
E' (complementary of E) = {11, 13, 15, 17}

Problem 6: Consider the experiment of tossing a fair coin 3 times.. Event A is defined as getting all tails. What kind of event is this? 

Solution: 

Sample space for the coin toss will be, 

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

For the event A, 
A = {TTT}

This event is only mapped to one element of sample space. Thus, it is a simple event. 

Problem 7: Let's say a coin is tossed once, State whether the following statement is True or False. 

“If we define an event X, which means getting both heads and tails. This event will be a simple event.”

Solution:

When a coin it tossed, there can be only two outcomes, Heads or Tails. 
S = {H, T} 

Getting both Heads and Tails is not possible, thus event X is an empty set. 
Thus, it is an impossible and sure event. So, this statement is False. 

Problem 8: A die is rolled, and three events A, B, and C are defined below:

• A: Getting a number greater than 3 
• B: Getting a number that is a multiple of 3. 
• C: Getting an odd number

Find A ∩ B, A ∩ B ∩ C, and A ∪ B.

Solution:

Sample space for die roll will be, 
S = {1, 2, 3, 4, 5, 6} 

For the event A, 
A = {4, 5, 6}

For the event B, 

B = {3, 6}

For the event C, 
C = {1, 3, 5}

A ∩ B = {4, 5, 6} ∩ {3, 6}
A ∩ B = {6}

A ∩ B ∩ C = {4, 5, 6} ∩ {3, 6} ∩ {1, 3, 5}
A ∩ B ∩ C = ∅ (Empty Set) 

A ∪ B = {4, 5, 6} ∪ {3, 6}
A ∪ B = {3, 4, 5, 6}

Problem 9: A die is rolled. Let's define two events: Event A is getting the number 2, and Event B is getting an even number. Are these events mutually exclusive? 

Solution: 

Sample space for die roll will be, S = {1, 2, 3, 4, 5, 6} 

For the event A, 
A = {2}

For the event B, 
B = {2, 4, 6}

For two events to be mutually exclusive, their intersection must be an empty set 

A ∩ B = {2} ∩ {2, 4, 6}
A ∩ B  = {2}

Since it is not an empty set, these events are not mutually exclusive.

Outcome: Outcome is a single result that can occur from an experiment. 
Event: An event is a collection of outcomes that share a common characteristic.

What are the Types of Events in Probability?

The different types of events in probability are as follows:

• Impossible and Sure Events
• Simple Event and Compound Event
• Dependent and Independent Events
• Mutually Exclusive Events
• Exhaustive Events
• Equally Likely Events

What is a Simple Event?

A simple event is an event that consists of a single outcome. Example: if a coin is tossed, the event of getting a head is a simple event.

What is a Compound Event?

A compound event is an event that consists of two or more outcomes. Example: when rolling two dice and getting a sum of 7 which can be achieved either by (1, 6), (2, 5), (3, 4), (4, 3), (3, 4), (5, 2), or (6, 1).

What is the Difference Between a Simple Event and a Compound Event in Probability?

Simple Event: A simple event in probability is an event that include only a single outcome. 
Compound Event: A compound event is an event that consists of two or more outcomes.

What is the Complement of an Event in Probability?

The complement of an event in probability is the set of all outcomes in the sample space that are not in the event.

What is the Intersection of Two Events in Probability?

The intersection of two events is the event that consists of all outcomes that are in both events. It is denoted by the symbol ∩.

What is the Union of Two Events in Probability?

The union of two events is the event that consists of all outcomes that are in either of the two events. It is denoted by the symbol ∪.