If you are a Mathematics or Statistics student, we are sure you have come across the concept of Probability, right? Yes, we know, the idea is kind of complex and frustrating, but extremely important too.
In reality, Probability refers to the chances and likelihood of something going to happen or an event occurring. We come across trillions of uncertain situations in our lives where we are not sure what will happen. When this happens, you are not sure about the Probability of an event occurring.
Have you ever heard someone saying, "There are chances of snowfall tomorrow."? Or someone saying, "I think I will score 80% in my exam this year."? Yes, these are a few common examples of Probability.
But the question is, what is Probability in Mathematics and Statistics, and how do you calculate Probability? Well, you can calculate it with the help of the probability formula! In this post, we will explain the probability formula, its definition, and more basics you need to know. Keep Reading!
Definition of Probability
But in Mathematics and Statistics, Probability is the ratio of the total number of possible outcomes and the total number of outcomes of an event occurring. The event is represented as E, and Probability is represented as P. In simpler words:
Probability = Possible outcomes/Total outcomes
Applications of Probability in Real Life
We use Probability in our day-to-day lives without even knowing it. Here are a few examples of how:
- Choosing the card from the deck while playing.
- Flipping coins before a match or a game.
- Throwing dice while playing board games.
- Lucky draws.
- Weather Predictions.
Terms in the Probability Theory
Some specific terms help students to understand the concept of Probability more clearly. Let's take a look at them:
Experiment
An experiment in Probability is a trial operation performed to find an outcome.
Sample Space
Sample space refers to all the favourable and possible outcomes expected after an experiment. For example, if our experiment is tossing a coin, our sample space would be tail and head.
Favourable Outcomes
Favourable outcomes refer to getting our desired results after an event and experiment. For example, if you are rolling a dice and you want the result to be 5, the favourable outcomes will be (3,2), (4,1). (2,3), and (1,4).
Trial
A trial refers to performing a random experiment to see the outcomes of an event happening.
Random Experiment
A random experiment is a trial where we don't know about the results we will get. For example, if we are tossing a coin, we know we will get a head or a tail, but we are not sure which one.
Event
Event is the total number of outcomes we get after performing a random experiment.
Exhaustive Events
An exhaustive event happens when all the outcomes after an experiment are equal to the sample space.
Mutually Exclusive Events
Mutually exclusive events are events that can not happen together. For example, it can either be spring or autumn; we can not see both of these climates together. Therefore, spring and autumn are two mutually exclusive events.
Probability Formula
The formula to calculate the probability of an event is:
P(A) = Number of Favorable Outcomes / Total Number of Possible Outcomes
In other words,
P(A) = n(A)/n(S)
In this probability formula,
P(A) = Probability of an event occurring (A)
n(A) = number of favorable outcomes
n(S) = total number of events in the sample space
Basic Probability Formulas
If A and B are two events, these will be their basic probability formulas:
- Probability Range = 0 ≤ P(A) ≤ 1
- Addition Rule = P(A∪B) = P(A) + P(B) – P(A∩B)
- Rule of Complementary Events = P(A’) + P(A) = 1
- Disjoint Events = P(A∩B) = 0
- Independent Events Probability Formula = P(A∩B) = P(A) ⋅ P(B)
- Conditional Probability Formula = P(A | B) = P(A∩B) / P(B)
- Bayes Formula = P(A | B) = P(B | A) ⋅ P(A) / P(B)
Probability Tree Diagram
Now that you have gone through the probability formulas, it's time to go through the probability tree diagram! This visual representation helps to find the possible outcomes of a specific event happening or not happening.
Below, we are sharing a tree diagram of a supposed experiment of tossing a coin to help students understand Probability and possible outcomes.
Types of Probability
There are several types of Probability you can come across while finding the Probability of an event occurring. Let's go through these main types in detail now!
Classical Probability
Classical Probability, also known as theoretical Probability, refers to the possible chances of something going to happen. Moreover, classical Probability helps find the reasoning behind the real Probability. For example, if we are tossing a coin, we have a 1/2 theoretical probability of getting a tail.
Empirical Probability
Empirical Probability, also known as experimental Probability, works by focusing on the observations of thought experiments. You can calculate the experimental Probability by dividing the total number of trials with possible outcomes. For example, if you toss a coin 12 times and you get the head 5 times, then the empirical Probability will be 5/12.
Subjective Probability
Subjective Probability is something that is not based on any math formulas or calculations. Instead, it is based on a person's own beliefs and thinking about a certain event happening. For example, in a cricket match between Australia and India, a person thinks Australia is going to win; that is subjective Probability.
Axiomatic Probability
Axiomatic Probability works by following the 3 sets of Axioms rules by Kolmogorov. All the chances of happening or not happening of an event can be determined by applying the following rules:
- The smallest number of possible probabilities will always be 0, and the largest will be 1.
- The Probability of a certain event will always be equal to 1.
- Two mutually exclusive events can never occur together at the same time. If the first event is occurring, the second event can never occur at the same time.
How to Find the Probability of an Event?
Now that we have gone through all the teeny tiny and essential details, it's time to see how can we find the Probability of an event.
Now we already know that the Probability of an event means the possibility of that event, right? Also, the value of the Probability can only range from 0 to 1.
Suppose, an event E is going to happen, we will represent it as P(E). Now, figuring out the probabilities in the following ways:
P(E) = 0 (Only if E event is not occurring)
P(E) = 1 (Only if we are certain E is going to occur)
0 ≤ P(E) ≤ 1.
Theorems of Probability
Like every other theory in Mathematics, Probability also has some basic theorems for a better understanding of the concept and its applications. These theorems are what help to perform the numerical calculations in Probability. Let's take a look at them!
Theorem # 1: The sum of the probability of an event happening and not will always be equal to 1.
P(A)+P(¯A)=1
Theorem # 2: The Probability of an event not happening or can't happen will always be equal to 0.
P(ϕ)=0
Theorem # 3: The Probability of an event definitely happening will always be equal to 1.
P(A) = 1
Theorem # 4: The Probability of if an event is going to happen or not will always lie between 0 and 1.
0 < P(A) < 1
Theorem # 5: If there are two events, A and B, and they are not joined, we can apply the union formula of 2 sets to them.
P(A∪B)=P(A)+P(B)−P(A∩B)
Bayes' Theorem of Conditional Probability
Remember when we read about conditional Probability above? That is what this theorem is about. It helps to calculate the possibility and Probability of one event based on the observations of happening of another event.
For example, we have 3 sacks, and each bag has purple, blue, and green balls. So, what is the probability that we will get a green ball from the third sack? Because we have different colored balls in other sacks. This type of event is called conditional Probability.
The probability formula of Bayes' Theorem is:
P(A|B) = P(B|A).P(A)/P(B)
Here,
P(A) = possibility that event A is going to occur.
P(B) = possibility that event B is going to occur.
Law of Total Probability
If n is the number of events while experimenting, then the sum of all event probabilities will be equal to 1.
P(A1) + P(A2) + P(A3) +. ... P(An) = 1
Solved Examples of Probability
Example 1: If you throw two dice in the air, what is the probability of getting a sum of 10?
Solution:
Two dice can give us a total of 36 possible outcomes. Since our desired outcome is 10, these are the favourable outcomes we can get:
(5, 5), (4, 6), and (6, 4).
Now, Probability is the number of favourable outcomes divided by total outcomes in sample space. Hence,
P(E) = 3/36 = 1/12
Therefore, the probability of getting a sum of 10 is 1/12.
Example 2: Find the Probability of getting 5 on rolling dice.
Solution:
The sample space of one rolling dice will be (1, 2, 3, 4, 5, 6)
Hence, the number of total outcomes = 6
Number of favorable outcomes = 1
Now, P(A) = n(A)/n(S) = ⅙
Therefore, the probability of getting 5 on a rolling dice is 1/6.
Example 3: You need to get and draw out a face card from a deck of cards. What is the probability of this going to happen?
Solution:
A normal deck has 52 cards. Therefore,
Total number of outcomes = 52
Now, the face cards are 3 (King, Queen, and Jack). Hence,
Number of favorable events N(E) = 4 x 3 = 12
Now, we apply the probability formula:
Probability = (Favorable Outcomes) ÷ (Total Favorable Outcomes)
Probability = 12 ÷ 52
P = 3/13
Therefore, the probability of drawing a face card is 3/13.
Example 4: What is the probability of getting an odd number if you roll one dice?
Solution:
The sample space of 1 dice = (1, 2, 3, 4, 5, 6)
n(S) = 6
Now, the number of possibilities of the event = (1, 3, 5)
n(E) = 3
Now, we apply the formula:
= n(E)/n(S)
= 3/6
= 1/2
Frequently Asked Questions
What is Probability?
Probability is an interesting branch of Mathematics that finds out the possibility of an event is going to happen or not. The value of Probability can range from 0 to 1. If it is 1, then the probability of an event is certain, and if it is 0, then the Probability is uncertain.
What is the probability formula?
The probability formula is expressed in the following way:
Probability of an event P(E) = (Number of favorable outcomes) ÷ (Sample space).
What are the main types of Probability?
The main types of Probability are as follows:
Theoretical Probability / Classical Probability
Experimental Probability / Empirical Probability
Axiomatic Probability
When was Probability discovered?
Probability was discovered back in the seventeenth century when the word probable emerged. At that time, this word was referred to things said and done by sensible and wise people. After some time, the word was referred to as something legal with solid proof. As time went by, the actual concept of Probability came into being through a lot of research, theories, and study.
Give some real-life examples of Probability that we use in our daily lives?
We use the concept of Probability in the following ways in our daily lives:
- Weather Prediction.
- Card Games.
- Political Voting.
- Dice Games.
- Predicting who is going to win in sports games.
Final Words
Probability is indeed an interesting topic, but only if you learn it the right way. You might find it hard and confuse at the start, but trust me; once you understand Probability, you will be able to solve every problem and question like a PRO!
We hope that our little guide helped you understand Probability. Still, if you have any more questions and confusion, you can always request us for a professional
tutor. Our highly experienced tutors will make everything super easy for you. We are just a click away!
Find Top Tutors in Your Area
Find A Tutor
