Probability theory is an advanced branch of mathematics that deals with measuring the likelihood of events occurring. It provides tools to analyze situations involving uncertainty and helps in determining how likely certain outcomes are. This theory uses the concepts of random variables, sample space, probability distributions, and more to determine the outcome of any situation.
For Example: Flipping a Coin
Flipping a coin is a random event with two possible outcomes: heads or tails. Each time you flip a fair coin, there are exactly two possible outcomes, each with an equal chance of occurring. Therefore, the probability of landing on heads is 1/2, and similarly, the probability of landing on tails is also 1/2.
Different Approaches In Probability Theory
Probability theory studies random events and tells us about their occurrence. The three main approaches for studying probability theory are:
Theoretical Probability
Theoretical Probability deals with assumptions to avoid unfeasible or costly repetition of experiments. The theoretical Probability for an event A can be calculated as follows:
P(A) = (Number of outcomes favorable to Event A) / (Number of all possible outcomes)
The image shown below shows the theoretical probability formula.
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Note: Here we assume the outcomes of an event as equally likely.
Now, as we learn the formula, let’s put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail. Hence, The Probability of occurrence of Head on tossing a coin is P(H) = 1/2
Similarly, The Probability of the occurrence of a Tail on tossing a coin is P(T) = 1/2
Experimental Probability
Experimental probability is found by performing a series of experiments and observing their outcomes. These random experiments are also known as trials. The experimental probability for Event A can be calculated as follows:
P(E) = (Number of times event A happened) / (Total number of trials)
The following image shows the Experimental Probability Formula,
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Now, as we learn the formula, let’s put this formula in our coin-tossing case. If we tossed a coin 10 times and recorded heads for 4 times and a tail 6 times then the Probability of occurrence of Head on tossing a coin: P(H) = 4/10
Similarly, the Probability of Occurrence of Tails on tossing a coin: P(T) = 6/10
Subjective Probability
Subjective probability refers to the likelihood of an event occurring, as estimated by an individual based on their personal beliefs, experiences, intuition, or knowledge, rather than on objective statistical data or formal mathematical models.
Example: A cricket enthusiast might assign a 70% probability to a team’s victory based on their understanding of the team’s recent form, the opponent’s strengths and weaknesses, and other relevant factors.
Basics of Probability Theory
Random Experiment
In probability theory, any event that can be repeated multiple times and its outcome is not hampered by its repetition is called a Random Experiment. Tossing a coin, rolling dice, etc. are random experiments.
Sample Space
The set of all possible outcomes for any random experiment is called sample space. For example, throwing dice results in six outcomes, which are 1, 2, 3, 4, 5, and 6. Thus, its sample space is (1, 2, 3, 4, 5, 6)
Event
The outcome of any experiment is called an event. Various types of events used in probability theory are,
- Independent Events: The events whose outcomes are not affected by the outcomes of other future and/or past events are called independent events. For example, the output of tossing a coin in repetition is not affected by its previous outcome.
- Dependent Events: The events whose outcomes are affected by the outcome of other events are called dependent events. For example, picking oranges from a bag that contains 100 oranges without replacement.
- Mutually Exclusive Events: The events that can not occur simultaneously are called mutually exclusive events. For example, obtaining a head or a tail in tossing a coin, because both (head and tail) can not be obtained together.
- Equally likely Events: The events that have an equal chance or probability of happening are known as equally likely events. For example, observing any face in rolling dice has an equal probability of 1/6.
Random Variable
A variable that can assume the value of all possible outcomes of an experiment is called a random variable in Probability Theory. Random variables in probability theory are of two types which are discussed below,
Discrete Random Variable
Variables that can take countable values such as 0, 1, 2,… are called discrete random variables.
Continuous Random Variable
Variables that can take an infinite number of values in a given range are called continuous random variables.
Various formulas are used in probability theory and some of them are discussed below,
- Theoretical Probability Formula: (Number of Favourable Outcomes) / (Number of Total Outcomes)
- Empirical Probability Formula: (Number of times event A happened) / (Total number of trials)
- Addition Rule of Probability: P(A ∪ B) = P(A) + P(B) – P(A∩B)
- Complementary Rule of Probability: P(A’) = 1 – P(A)
- Independent Events: P(A∩B) = P(A) ⋅ P(B)
- Conditional Probability: P(A | B) = P(A∩B) / P(B)
- Bayes’ Theorem: P(A | B) = P(B | A) ⋅ P(A) / P(B)
Read in Detail: Bayes Theorem
Probability Theory in Statistics
Probability has various applications in Statistics. These are mentioned below:
- Descriptive Statistics: Probability theory helps in understanding and interpreting data summaries and distributions.
- Inferential Statistics: This forms the basis for making inferences about populations from samples, including hypothesis testing and the construction of confidence intervals.
- Regression Analysis: Probability distributions of errors are used to estimate the relationships between variables.
- Bayesian Statistics: Uses probability to represent uncertainty about the parameters of interest and updates this uncertainty as more data becomes available.
Applications of Probability Theory
Probability theory is widely used in our lives, it is used to find answers to various types of questions, such as Will it rain tomorrow? What is the chance of landing on the Moon? What is the chance of the evolution of humans? And others. Some of the important uses of probability theory are,
- Probability theory is used to predict the performance of stocks and bonds.
- In casinos and gambling probability theory is used to find the chances of winning.
- Probability theory is used in weather forecasting.
- Probability theory is used in Risk mitigation.
- In consumer industries, the risk of product failure is mitigated by using the Probability theory.
Solved Examples of Probability Theory
We can study the concept of probability with the help of the example discussed below,
Example 1: Let’s take two random dice and roll them randomly, now the probability of getting a total of 10 is calculated.
Solution:
Total Possible events that can occur (sample space) {(1,1), (1,2),…, (1,6),…, (6,6)}. The total spaces are 36.
Now the required events, {(4,6), (5,5), (6,4)} are all which adds up to 10.
So the probability of getting a total of 10 is = 3/36 = 1/12
Example 2: A fair coin is tossed three times. What is the probability of getting exactly two heads?
Solution:
Total possible outcomes when tossing a coin three times = 23 = 8.
Possible outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT.
Outcomes with exactly two heads: HHT, HTH, THH (3 outcomes).
Probability of getting exactly two heads:
P(exactly 2 heads)=Number of favorable outcomes/ Total outcomes.
P(exactly 2 heads)=3/ 8.
Example 3: A standard deck of cards contains 52 cards. What is the probability of drawing an Ace or a King from the deck?
Solution:
Total number of cards = 52.
Number of Aces = 4.
Number of Kings = 4.
Total number of favorable outcomes (Aces or Kings) = 4 + 4 = 8.
Probability of drawing an Ace or a King:
P(Ace or King)=Number of favorable outcomes/Total outcomes
P(Ace or King)=Number of Aces or Kings/Total number of cards.
P(Ace or King) = 8/52 = 2/13.
Example 4: Consider a jar with 7 red marbles, 3 green marbles, and 4 blue marbles. What is the probability of randomly selecting a non-blue marble from the jar?
Solution:
Given,
Number of Red Marbles = 7, Number of Green Marbles = 3, Number of Blue Marbles = 4
So, Total number of possible outcomes in this case: 7 + 3 + 4 = 14
Now, Number of non-blue marbles are: 7 + 3 = 10
According to the formula of theoretical Probability we can find, P(Non-Blue) = 10/14 = 5/7
Hence, theoretical probability of selecting a non-blue marble is 5/7.
Example 5: Consider Two players, Naveena and Isha, playing a table tennis match. The probability of Naveena winning the match is 0.76. What is the probability of Isha winning the match?
Solution:
Let N and M represent the events that Naveena wins the match and Isha wins the match, respectively.
The probability of Naveena’s winning P(N) = 0.62 (given)
The probability of Isha’s winning P(I) = ?
Winning of the match is an mutually exclusive event, since only one of them can win the match.
Therefore,
P(N) + P(I) =1
P(I) = 1 – P(N)
P(I) = 1 – 0.62 = 0.38
Thus, the Probability of Isha winning the match is 0.38.
Example 6: If someone takes out one card from a 52-card deck, what is the probability of the card being a heart? What is the probability of obtaining a 7-number card?
Solution:
Total number of cards in a deck = 52
Total Number of heart cards in a deck = 13
So, the probability of obtaining a heart,
P(heart) = 13/52 = 1/4
Total number of 7-number cards in a deck = 4
So, the probability of obtaining a 7-number card,
P(7-number) = 4/52 = 1/13
Example 7: Find the probability of rolling an even number when you roll a die containing the numbers 1-6. Express the probability as a fraction, decimal, ratio, or percent.
Solution:
Out of 1 to 6 number, even numbers are 2, 4, and 6.
So, Number of favorable outcomes = 3.
Total number of outcomes = 6.
Probability of obtaining an even number P(Even)= 1/2 = 0.5 = 1 : 2 = 50%
People Also Read:
Probability Theory – FAQs
What is the concept of Probability Theory?
The branch of mathematics that deals with the odds of happening an event is called the Probability Theory. It tells us about the odds of occurring an event and also all the possible outcomes of any event.
What are the two types of probabilities in Probability Theory?
The two types of probabilities in probability theory are
- Theoretical Probability
- Experimental probability
Who invented probability theory?
The credit for the creation of modern probability theory goes to Intalian mathematics Gerolamo Cardano.
What is a random variable in Probability Theory?
A variable that describes all the possible outcomes of any random experiment is called a random variable. A random variable can either be continuous or discrete.
The main formulas used in probability theory are,
- Theoretical probability: (Number of outcomes favourable to Event A) / (Number of all possible outcomes)
- Experimental Probability: (Number of times event A happened) / (Total number of trials)
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When two or more two solids are combined and the combination comes out useful, a shape that can be found in reality is called a combination of solids. When Solids are taught, the major focus is always on the point of their real-life use and applications, For example, a cylinder can be seen in Pipes
9 min read
Frustum of Cone
Frustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to cir
10 min read
Conversion of solids - Surface Areas and Volumes
Conversions or changes are now a normal feature of our everyday lives. A goldsmith, for instance, melts a strip of gold to turn it into a gem. Likewise, a kid plays with clay forms it into various toys, a carpenter uses the wooden logs to shape various items/furniture for housekeeping. Likewise, for
4 min read
Surface Areas and Volumes
Surface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces. Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate th
10 min read
Chapter 14: Statistics
How to find Mean of grouped data by direct method?
Statistics involves gathering, organizing, analyzing, interpreting, and presenting data to form opinions and make decisions. Applications range from educators computing average student scores and government officials conducting censuses to demographic analysis. Understanding and utilizing statistica
9 min read
Shortcut Method for Arithmetic Mean
Statistics, in layman's words, is the process of gathering, classifying, examining, interpreting, and finally presenting information in an understandable manner so that one can form an opinion and, if necessary, take action. Examples: A teacher collects students' grades, organizes them in ascending
6 min read
How to Calculate Mean using Step Deviation Method?
Step Deviation Method is a simplified way to calculate the mean of a grouped frequency distribution, especially when the class intervals are uniform. In simple words, statistics implies the process of gathering, sorting, examining, interpreting and then understandably presenting the data to enable o
7 min read
Graphical determination of Median
A measure of central tendency that determines the centrally located value of a given series is known as the Median. The number of values of the series below and above the given series is always equal. To determine the median value of a given series, it is first managed in increasing or decreasing or
5 min read
Ogive (Cumulative Frequency Curve) and its Types
A method of presenting data in the form of graphs that provides a quick and easier way to understand the trends of the given set of data is known as Graphic Presentation. The two types of graphs through which a given set of data can be presented are Frequency Distribution Graphs and Time Series Grap
6 min read
Chapter 15: Probability
Types of Events in Probability
Whenever an experiment is performed whose outcomes cannot be predicted with certainty, it is called a random experiment. In such cases, we can only measure which of the events is more likely or less likely to happen. This likelihood of events is measured in terms of probability and events refer to t
13 min read
Dependent and Independent Events
Dependent and Independent Events are the types of events that occur in probability. Suppose we have two events say Event A and Event B then if Event A and Event B are dependent events then the occurrence of one event is dependent on the occurrence of other events if they are independent events then
8 min read
Experimental Probability
Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probabil
8 min read
Probability Theory
Probability theory is an advanced branch of mathematics that deals with measuring the likelihood of events occurring. It provides tools to analyze situations involving uncertainty and helps in determining how likely certain outcomes are. This theory uses the concepts of random variables, sample spac
10 min read
Experimental Probability
Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probabil
8 min read
CBSE Class 10th Maths Formulas: Chapter Wise Formula and Points
Mathematics is one of the most scoring subject in CBSE Class 10th board exam. So Students are advised to prepare well for Math in order to score good marks in CBSE Class 10 board exam. GeeksforGeeks has curated the chapter wise Math formulae for CBSE Class 10th exam. These Formulae include chapters
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NCERT Solutions for Class 10 Maths 2024-25: Chapter Wise PDF Download
NCERT Solutions for Class 10 Maths are tailored by subject matter experts to assist 10 students in securing top marks in their 10 board exams. As we know mathematics is the highest-scoring subject in the CBSE Class 10th board exam This complete resource includes all questions and answers from the CB
15+ min read
RD Sharma Class 10 Solutions
RD Sharma Class 10 Solutions offer excellent reference material for students, enabling them to develop a firm understanding of the concepts covered. in each chapter of the textbook. As Class 10 mathematics is categorized into various crucial topics such as Algebra, Geometry, and Trigonometry, which
9 min read