Events in Probability

In probability, events represent the possible outcomes of a random experiment. Classifying events based on probability helps simplify calculations and improve understanding of how likely a result is to occur. The probability of an event can be calculated using the formula: 

\({\text {Probability of an event }}= {{{\text{number of favorable outcomes}} \over {\text {total number of possible outcomes}}}}\).

Here, the total number of possible outcomes of an experiment is the sample size, representing all outcomes that can occur during the experiment. 

For example, consider the experiment of flipping three fair coins. The sample space, the set of all possible outcomes, is 8, which are: 
 

Sample space = HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Sample size = 8

Let us consider event A to be getting at least two heads and event B to be getting at least one tail.

So, Event A = {HHH, HHT, HTH, THH}

Event B = {HHT, HTH, HTT, THH, THT, TTH, TTT}

The probability of event A: P(A) \({{4 \over 8 }}= {{1 \over 2}}\)

The probability of event B: P(B) \(= {{7\over 8}}\)

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. 

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.

      
\({\text{Probability (P)}} = {{{\text {Favorable Outcomes }}\over {\text {Total Outcomes}}}}\)

Step 4: The probability after applying the formula should be between 0 and 1.

For example, a box has six balls: 4 green balls and two yellow balls. What is the probability of picking a green ball at random?

Here, the total number of balls = 6

So, the sample space has six possible outcomes

The number of green balls = 4

So, the probability of getting green balls can be calculated using the formula: 

P(green balls) = \(4\over 6 \)
\(= {2 \over 3} \\ \ \\ = 0.667\)

So, the probability of getting green balls is 0.66 or 66%

Understanding probability helps students identify possible outcomes, make smart predictions, and solve problems related to probability. In this section, we will learn a few tips and tricks to master events in probability. 

• List all the possible outcomes before solving a problem. This makes it easier for students to find the probability. 
   
• Students should understand the difference between types of events, such as independent and dependent events. 
   
• Use diagrams or tables like lists, charts, or tree diagrams to organize outcomes. 
   
• Teachers can let students experiment with coins, dice, spinners, or colored counters as practical activities to help students create a stronger understanding. 
   
• Parents can help the child practice predicting the outcomes in everyday situations by asking simple questions like “What do you think is the chance it will rain today?” or “If we toss a coin, what might happen?”
   

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.