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Last updated on October 7, 2025

Events in Probability

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In probability, an event is something that may occur when we conduct an experiment. It can be a single outcome or a set of possible outcomes resulting from a random experiment.

Events in Probability for US Students
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What are 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 EA = {HHH, HHT, HTH, THH}. Next, the probability of Event B getting at least one tail is EB = {HHT, HTH, THH}.

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What are the Types of Events in Probability?

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.
 

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How to Find the Probability of an Event?

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.

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Real-Life Applications of Events in Probability

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. 

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Common Mistakes and How to Avoid Them in Events in Probability

Making mistakes when calculating probability is a common occurrence, particularly when the students are unfamiliar or new to this concept. Here are five common mistakes that students might make and how to avoid them. 

Mistake 1

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Miscounting the Sample Space

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Students can miscount the total number of possible outcomes. Carefully list all the possible outcomes before calculating probability. Double-check your count to ensure no outcome is missing or repeated.

Mistake 2

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Confusing Mutually Exclusive with Independent Events

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Though the concepts look similar, they have certain differences. Remember that mutually exclusive events cannot happen together, while independent events do not affect each other. 

Mistake 3

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Forgetting to Convert Probability to a Proper Fraction or Decimal 

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After finding the favorable outcomes of an event, divide it by the total outcomes or sample space to get the probability. Ensure that probability values are between 0 and 1 (or expressed as percentages between 0% and 100%). 

Mistake 4

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Assuming Equal Probability for All Outcomes

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Some outcomes may be more likely than others. Carefully analyze the situation instead of assuming equal chances.

Mistake 5

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Ignoring the Complementary Rule

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If finding the probability of an event is difficult, use P(A’) = 1 − P(A) to find the probability of the event not happening.

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Solved Examples of Events in Probability

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Problem 1

A fair six-sided die is rolled. What is the probability of getting a 5?

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1/6 

Explanation

The sample space for rolling a die is {1, 2, 3, 4, 5, 6}. There is only one favorable outcome (rolling a 5) out of six possible outcomes.

Using the probability formula, 


P = Favorable Outcomes / Total Outcomes = 1/6 ≈ 0.167

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Problem 2

A bag contains 3 red, 4 blue, and 5 green balls. What is the probability of drawing a blue ball?

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0.333

Explanation

Total no. of balls = 3 + 4 + 5 = 12

Favorable outcomes (blue balls) = 4

P(blue balls) = 4/12 = 1/3 ≈ 0.333
 

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Problem 3

A coin is tossed twice. What is the probability of getting at least one head?

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0.75

Explanation

The sample space for two coin tosses is {HH, HT, TH, TT}.


Favorable outcomes (at least one head) = {HH, HT, TH}


Total possible outcomes = 4


P(at least one head) = 3/4 = 0.75

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Problem 4

If probability of Event A is 0.5, and the sample space is 6, what is the favorable outcome of Event A?

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3

Explanation

The probability of the Event A (P) = 0.5

Sample space or the total number of possible outcomes = 6

According to the probability formula,

P = Number of Favorable Outcomes / Total Number of Possible Outcomes

⇒ 0.5 = Number of Favorable Outcomes / 6

⇒ Number of Favorable Outcomes = 6 × 0.5 = 3

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FAQs on Events in Probability

1.What is the probability of a sure event?

The probability of a sure event is always 1.

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2.How do you calculate the probability of at least one event occurring?

If the probability of failing a test is 0.2, the probability of passing at least once in two attempts is: P (At least one pass) = 1 – P; (Fail both times) = 1 – (0.2 × 0.2) = 1 – 0.04 = 0.96.

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3.Can probability be greater than 1?

No, probability is always between 0 and 1. If the calculated probability is greater than 1, there is a mistake in the calculation.

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4.What is the probability of an impossible event?

The probability of an impossible event is 0.

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Jaipreet Kour Wazir

About the Author

Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref

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: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!

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