Events in Probability

In probability, events are the possible outcomes of a random experiment. This probability-based categorization of events helps in the simplification of mathematical calculations. The number of favorable results divided by the total number of outcomes of that experiment can be used to determine the probability of events occurring. 

For example, when you flip three fair coins, there are eight possible outcomes. That is, HHT, HHH, HTH, HTT, THT, THH, TTH, and TTT. Let us define an event as A. Then, the probability of Event A getting at least two heads is E_A = {HHH, HHT, HTH, THH}. Next, the probability of Event B getting at least one tail is E_B = {HHT, HTH, THH}.

Events in probability can be classified into a variety of categories. A random experiment can only have one sample space (set of all possible outcomes of an experiment), but it can have a wide variety of events. 

For example, rolling a die will have only one sample space: 
 

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

But it can have different events: 
 

Event A of rolling an even number: (A) = {2, 4, 6}
 

Event B of rolling a number greater than 4: (B) = {5, 6}
 

Event C of rolling a 3: (C) = {3}
 

The following is a list of some important probability events.
 

• Independent and Dependent Events
   
• Impossible and Sure Events
   
• Complementary Events
   
• Mutually Exclusive Events
   
• Exhaustive Events
   
• Equally Likely Events

Independent and Dependent Events

In probability, independent events are events whose outcomes are not affected by the outcome of any previous event. For example, tossing a coin is an independent event, because the previous event like getting heads will not affect the next outcome. 

Dependent events are those events that will depend on the outcome of the previous results. For example, imagine you pick a ball, let’s say Ball A, from a bag containing different colors of balls. When you pick another ball, there won't be Ball A in the bag. That means the probability of not getting Ball A is already determined. 

Impossible and Sure Events

An event that will not happen is known as an impossible event. The probability of an impossible event is 0 (zero). For example, rolling a die numbered 7 is an impossible event because a die has numbers only from 1 to 6.

On the other hand, a sure event is the one that will happen for sure. The probability of an event that will happen for sure is 1 (one). For example, a sure event is the Sun rising tomorrow. It will happen no matter what (unless we consider extreme cosmic events). 

Simple and Compound Events

A simple event is when there is only one specific outcome out of all possible outcomes. For example, when rolling a six-sided die, the sample space (all possible outcomes) is {1, 2, 3, 4, 5, 6}. Getting a 4 on a rolling die refers to just one outcome 4, that is, E = {4}.
 

Whereas, an event that consists of more than one single event from the sample space is called a compound event. For example, getting an odd number on a die is a compound event as the events are E = {1, 3, 5} (multiple events from a single sample space).

Complementary Events

Complementary events are two events in which one of the two can only occur if and only if the other does not exist. The sum of the complementary events is 1 (one). For example, Event A of drawing a red ball from a bag is mutually exclusive with Event B of not drawing a red ball from the bag. This can be termed as Event A = E and Event B = E'. Then E and E' are complementary to each other.
 

Mutually Exclusive Events

Mutually exclusive events are those events that will not happen together. They do not have any common outcome. For example, Event A of rolling a die of number 4 is E = {4}, and Event B of rolling a die of number 3 is E = {3}. These are mutually exclusive because both Event A and Event B cannot occur at the same time.

Exhaustive Events

Exhaustive events are those events that cover all possible outcomes of an experiment. This means that during an experiment, at least one of these events must occur. For example, in an examination, the possible outcomes are passing or failing an exam.

Equally Likely Events

Events with equally conceivable outcomes are ones that have an equal likelihood of occurring. For example, tossing a coin has a 50% chance of getting heads and 50% chance of getting tails.
 

We can find the probability of events using four simple steps:

Step 1: First, we need to identify the sample space (list of all possible outcomes of the experiment).

Step 2: Decide what event you want to find and find how many outcomes match the event you are looking for. 

Step 3: Divide the number of favorable outcomes by the total number of possible outcomes.

      
Probability (P) = Favorable Outcomes / Total Outcomes

Step 4: The probability after applying the formula should be between 0 and 1. That is, for example, the probability of getting a 6 on a die is 

P = 1/6 = 0.167.

Here are some of the real-life applications of events in probability. Let’s understand them in detail:

Weather Forecasting

Probability is used to predict the chances of rain, storm, thunder, etc. 

Sports and Gaming

Probability is used in predicting match outcomes, player performance, and betting odds. 

Medical Field

Doctors use probability to assess disease risks and treatment’s success rate. 

Insurance and Risk Assessment

Insurance companies calculate accident or illness probabilities.