Event in Probability

Probability is the mathematics of uncertainty, and an event is the heart and soul of this uncertainty. It is the outcome or a group of outcomes representing our interest, whether it is calculating the chances of drawing a red card from a deck or predicting the result of a coin toss. Today, our topic of discussion will be this crucial concept of probability, “the Events”, its types and their practical applications.

1.0Event Definition in Probability

An event in probability reflects a collection of one or more outcomes, also known as the subsets, of a random experiment. In other words, an event is what we are trying to measure the probability of during an experiment. Some examples of events include coin tossing, rolling a die, and drawing a card from a deck. 

When studying events in probability, two concepts of events form the basis for the same, which are: 

• Sample Space: The sample is the set of all the possible results of that random experiment and is denoted with the letter “S”. For example, while tossing a coin, the sample space will be {H, T}, representing all the outcomes possible. 
• • In the context of the events, an event is a subset of the sample space S, that is: 

$E\subseteq S$

• Required (Favourable) Outcome: A required outcome is the condition that satisfies the condition of the event. Simply put, these are specific results from the sample space that we are interested in. For example, while tossing, say you want to get a Head, then this is the favourable or required outcome of that event. 

2.0Types of Events in Probability

In probability, not every event behaves the same way; that is, some are guaranteed to happen, while some can never happen. Owing to the differences between different events, events are classified into various types, which are: 

Simple (Elementary) Event

An event that can not be further divided and consists of only one outcome is known as a simple event. In simple words, an elementary event is an event which has only one outcome from the sample space. For example, when rolling a die, 5 can be obtained only once. Note that every outcome in a sample space is a simple event.  

Compound (or Composite) Event

As the name suggests, an event that has multiple outcomes, in probability, such an event is classified as a compound event. It has two or more outcomes from the sample space. For example, getting an even number (2, 4, or 6) when rolling a die has more than one outcome.

Sure (or Certain) Event

An event that has a 100% guarantee to happen is known as a sure event. In simple words, it is an event that will happen in every case in an experiment. For example, sunset and sunrise every day are certain events that have a 100% guarantee to happen. Note that the probability of such events is always one. 

Impossible Event

An impossible event is completely the opposite of a sure event, meaning an event that cannot happen under any condition is known as an impossible event. For example, getting a 7 on a standard six-faced die is not possible. The probability of such events is always 0. 

Mutually Exclusive Events

Two or more events that can not happen at the same time are known as mutually exclusive events. In other words, if the occurrence of one event means the other event cannot occur, then the events are mutually exclusive. For example, getting an even and odd number at the same time from the natural numbers is not possible.  

Exhaustive Events

When two or more events cover all the possible outcomes of a sample space, then such events are termed as exhaustive events. For example, when tossing a coin, event 1 covers getting a head, while event 2 covers getting a tail; then events 1 and 2 are exhaustive. 

Independent and Dependent Events

Two events are independent when the probability of one event doesn’t affect the probability of another event. While in dependent events, the probability of one event affects the probability of another event. That is, we can say that any two events can either be independent or dependent. 

Note: For independent events A and B, the probability has the following relation: P(A and B)=P(A)P(B)

3.0Operations on Events

Just like any other subset, operations can be performed on events, too. Some major operations on events are: 

• Union of Events (A ∪ B): The union of events represent the occurrence of event A or event B or both. The formula for the operation is: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$
• Intersection of Events (A ∩ B): When both events, say A and B, occur together. For example, when rolling a die, getting a number that is both even and prime (2). 
• Complement of an Event (A'): The opposite of an event A is known as the complement of an event. The formula for the operation is: P(A')=1-P(A)
• Difference Between two Events (A-B): The difference between two events, say A and B, represent all the outcomes that are in A but not in B. 

4.0Probability of Events

The probability is the mathematical measure of the chances of an event happening. The formula to calculate the probability of events is expressed as: 

$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes in sample space}}$

Note: The probability of events can neither be negative nor more than one, that is $0\le P(E)\le 1.$ 

Examples: 

• The probability of getting a 2 when rolling a die is $\frac{1}{6}$.
• The probability of getting an even number when rolling a die is ½. 

5.0Solved Examples

Problem 1: A card is drawn at random from a standard deck of 52 cards. What is the probability of getting a red king or a queen? 

Solution: In a standard deck of cards, 

The number of red kings = 2 

The number of queens in each suit = 4

The probability of getting a red king or a queen = Probability of getting a red king + probability of getting a queen 

$$\begin{gathered} P(\text{Red king or Queen})=\frac{\text{the number of red kings}}{\text{Total number of cards}}+\frac{\text{The number of queens}}{\text{total number of cards}} \\ P(\text{Red king or Queen})=\frac{2}{52}+\frac{4}{52}=\frac{6}{52}=\frac{3}{26} \end{gathered}$$

Problem 2: In a class of 40 students, 28 passed Mathematics. If a student is selected at random, what is the probability that the student did not pass Mathematics? 

Solution: According to the question, 

Total number of students = 40 

The number of passed students = 28

$$\begin{gathered} \text{Probability of students who passed}=\frac{\text{The number of passed students}}{\text{Total number of students}} \\ P(\text{pass students})=\frac{28}{40}=\frac{7}{10} \\ \text{Hence, using the complement rule, the probability of students who didn’t pass is:}P(\text{not pass})=1-P(\text{pass})=1-\frac{7}{10}=\frac{3}{10} \end{gathered}$$

Problem 3: A bag contains 4 red, 5 green, and 3 blue balls. A ball is drawn at random. Given that the ball drawn is not blue, what is the probability that it is red?

Solution: Given that the total number of balls in a bag is 4 + 5 + 3 = 12

Total non-blue balls = 4+5 = 12 

Red balls among non-blue ones = 4 

$$\begin{gathered} \text{Probability of red and not blue ball}=\frac{\text{Red balls among non–blue ones}}{\text{Total non–blue balls}} \\ P(\text{Red}\mid \text{Not Blue})=\frac{4}{9} \end{gathered}$$