Bayes’ Theorem is a mathematical formula that helps determine the conditional probability of an event based on prior knowledge and new evidence.
It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.
Bayes’ Theorem helps us update probabilities based on prior knowledge and new evidence. In this case, knowing that the pet is quiet (new information), we can use Bayes’ Theorem to calculate the updated probability of the pet being a cat or a dog, based on how likely each animal is to be quiet.
Bayes Theorem and Conditional Probability
Bayes’ theorem (also known as the Bayes Rule or Bayes Law) is used to determine the conditional probability of event A when event B has already occurred.
The general statement of Bayes’ theorem is “The conditional probability of an event A, given the occurrence of another event B, is equal to the product of the event of B, given A, and the probability of A divided by the probability of event B.” i.e.
For example, if we want to find the probability that a white marble drawn at random came from the first bag, given that a white marble has already been drawn, and there are three bags each containing some white and black marbles, then we can use Bayes’ Theorem.
Check: Bayes’s Theorem for Conditional Probability
For any two events A and B, Bayes’s formula for the Bayes theorem is given by:
Formula for the Bayes theorem
Where,
- P(A) and P(B) are the probabilities of events A and B, also, P(B) is never equal to zero.
- P(A|B) is the probability of event A when event B happens,
- P(B|A) is the probability of event B when A happens.
Bayes Theorem Statement
Bayes’s Theorem for n sets of events is defined as,
Let E1, E2,…, En be a set of events associated with the sample space S, in which all the events E1, E2,…, En have a non-zero probability of occurrence. All the events E1, E2,…, E form a partition of S. Let A be an event from space S for which we have to find the probability, then according to Bayes theorem,
[Tex]P(E_i \mid A) = \frac{P(E_i) \cdot P(A \mid E_i)}{\sum_{k=1}^{n} P(E_k) \cdot P(A \mid E_k)}[/Tex]
for k = 1, 2, 3, …., n
Bayes Theorem Derivation
The proof of Bayes’s, Theorem is given as, according to the conditional probability formula,
[Tex]P(E_i \mid A) = \frac{P(E_i \cap A)}{P(A)}[/Tex]…..(i)
Then, by using the multiplication rule of probability, we get
[Tex]P(E_i \cap A) = P(E_i) \cdot P(A \mid E_i)[/Tex]……(ii)
Now, by the total probability theorem,
[Tex]P(A) = \sum_{k=1}^{n} P(E_k) \cdot P(A \mid E_k)[/Tex]…..(iii)
Substituting the value of P(Ei∩A) and P(A) from eq (ii) and eq(iii) in eq(i) we get,
[Tex]P(E_i \mid A) = \frac{P(E_i) \cdot P(A \mid E_i)}{\sum_{k=1}^{n} P(E_k) \cdot P(A \mid E_k)}[/Tex]
Bayes’ theorem is also known as the formula for the probability of “causes”. As we know, the Ei‘s are a partition of the sample space S, and at any given time, only one of the events Ei occurs. Thus, we conclude that the Bayes theorem formula gives the probability of a particular Ei, given that event A has occurred.
After learning about Bayes theorem in detail, let us understand some important terms related to the concepts we covered in the formula and derivation.
Hypotheses
- Hypotheses refer to possible events or outcomes in the sample space, they are denoted as E1, E2, …, En.
- Each hypothesis represents a distinct scenario that could explain an observed event.
Priori Probability
- Priori Probability P(Ei) is the initial probability of an event occurring before any new data is taken into account.
- It reflects existing knowledge or assumptions about the event.
- Example: The probability of a person having a disease before taking a test.
Posterior Probability
- Posterior probability (P(Ei∣A) is the updated probability of an event after considering new information.
- It is derived using the Bayes Theorem.
- Example: The probability of having a disease given a positive test result.
Conditional Probability
- The probability of an event A based on the occurrence of another event B is termed conditional Probability.
- It is denoted as P(A|B) and represents the probability of A when event B has already happened.
Joint Probability
- When the probability of two or more events occurring together and at the same time is measured, it is marked as Joint Probability.
- For two events A and B, it is denoted by joint probability is denoted as P(A∩B).
Random Variables
- Real-valued variables whose possible values are determined by random experiments are called random variables.
- The probability of finding such variables is the experimental probability.
Bayes Theorem Applications
Bayesian inference is very important and has found application in various activities, including medicine, science, philosophy, engineering, sports, law, etc., and Bayesian inference is directly derived from Bayes theorem.
Some of the Key Applications are:
- Medical Testing → Finding the real probability of having a disease after a positive test.
- Spam Filters → Checking if an email is spam based on keywords.
- Weather Prediction → Updating the chance of rain based on new data.
- AI & Machine Learning → Used in Naïve Bayes classifiers to predict outcomes.
Check, Bayes’ Life Applications of Bayes theorem
Difference Between Conditional Probability and Bayes Theorem
The difference between Conditional Probability and Bayes’s. The theorem can be understood with the help of the table given below.
| Bayes Theorem | Conditional Probability |
|---|
| Bayes’s Theorem is derived using the definition of conditional probability. It is used to find the reverse probability. | Conditional Probability is the probability of event A when event B has already occurred. |
| Formula: P(A|B) = [P(B|A)P(A)] / P(B) | Formula: P(A|B) = P(A∩B) / P(B) |
Purpose: To update the probability of an event based on new evidence.
| Purpose: To find the probability of one event based on the occurrence of another.
|
Focus: Uses prior knowledge and evidence to compute a revised probability.
| Focus: Direct relationship between two events.
|
Theorem of Total Probability
Let E1, E2,…., En be mutually exclusive and exhaustive events of a sample space S, and let E be any event that occurs with some Ei. Then, prove that :
P(E) = n∑i=1P(E/Ei) . P(Ei)
Proof:
Let S be the sample space.
Since the events E1, E2,…,En are mutually exclusive and exhaustive, we have:
S = E1 ∪ E2 ∪ E3 ∪ . . . ∪ En and Ei ∩ Ej = ∅ for i ≠ j.
Now, consider the event E: E = E ∩ S
Substituting S with the union of Ei’s:
⇒ E = E ∩ (E1 ∪ E2 ∪ E3 ∪ . . . ∪ En)
Using distributive law:
⇒ E = (E ∩ E1) ∪ (E ∩ E2) ∪ . . . ∪ (E ∩ En)
Since the events Ei are mutually exclusive, the intersections E∩Ei are also mutually exclusive. Therefore:
P(E) = P{(E ∩ E1) ∪ (E ∩ E2)∪ . . . ∪(E ∩ En)}
⇒ P(E) = P(E ∩ E1) + P(E ∩ E2) + . . . + P(E ∩ En)
{Therefore, (E ∩ E1), (E ∩ E2), . . . ,(E ∩ En)} are pairwise disjoint}
⇒ P(E) = P(E/E1) . P(E1) + P(E/E2) . P(E2) + . . . + P(E/En) . P(En) [by multiplication theorem]
⇒ P(E) = n∑i=1P(E/Ei) . P(Ei)
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Solved Examples of Bayes’s Theorem
Example 1: A person has undertaken a job. The probabilities of completion of the job on time with and without rain are 0.44 and 0.9, and 5, respectively. If the probability that it will rain is 0.45, then determine the probability that the job will be completed on time.
Solution:
Let E1 be the event that the mining job will be completed on time and E2 be the event that it rains. We have,
P(A) = 0.45,
P(no rain) = P(B) = 1 − P(A) = 1 − 0.45 = 0.55
By multiplication law of probability,
P(E1) = 0.44, and P(E2) = 0.95
Since, events A and B form partitions of the sample space S, by total probability theorem, we have
P(E) = P(A) P(E1) + P(B) P(E2)
⇒ P(E) = 0.45 × 0.44 + 0.55 × 0.95
⇒ P(E) = 0.198 + 0.5225 = 0.7205
So, the probability that the job will be completed on time is 0.7205
Example 2: There are three urns containing 3 white and 2 black balls, 2 white and 3 black balls, and 1 black and 4 white balls, respectively. There is an equal probability of each urn being chosen. One ball is equal probability chosen at random. What is the probability that a white ball will be drawn?
Solution:
Let E1, E2, and E3 be the events of choosing the first, second, and third urn respectively. Then,
P(E1) = P(E2) = P(E3) = 1/3
Let E be the event that a white ball is drawn. Then,
P(E/E1) = 3/5, P(E/E2) = 2/5, P(E/E3) = 4/5
By theorem of total probability, we have
P(E) = P(E/E1) . P(E1) + P(E/E2) . P(E2) + P(E/E3) . P(E3)
⇒ P(E) = (3/5 × 1/3) + (2/5 × 1/3) + (4/5 × 1/3)
⇒ P(E) = 9/15 = 3/5
Example 3: A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both hearts. Find the probability of the lost card being a heart.
Solution:
Let E1, E2, E3, and E4 be the events of losing a card of hearts, clubs, spades, and diamonds respectively.
Then P(E1) = P(E2) = P(E3) = P(E4) = 13/52 = 1/4.
Let E be the event of drawing 2 hearts from the remaining 51 cards. Then,
P(E|E1) = probability of drawing 2 hearts, given that a card of hearts is missing
⇒ P(E|E1) = 12C2 / 51C2 = (12 × 11)/2! × 2!/(51 × 50) = 22/425
P(E|E2) = probability of drawing 2 clubs ,given that a card of clubs is missing
⇒ P(E|E2) = 13C2 / 51C2 = (13 × 12)/2! × 2!/(51 × 50) = 26/425
P(E|E3) = probability of drawing 2 spades ,given that a card of hearts is missing
⇒ P(E|E3) = 13C2 / 51C2 = 26/425
P(E|E4) = probability of drawing 2 diamonds ,given that a card of diamonds is missing
⇒ P(E|E4) = 13C2 / 51C2 = 26/425
Therefore,
P(E1|E) = probability of the lost card is being a heart, given the 2 hearts are drawn from the remaining 51 cards
⇒ P(E1|E) = P(E1) . P(E|E1)/P(E1) . P(E|E1) + P(E2) . P(E|E2) + P(E3) . P(E|E3) + P(E4) . P(E|E4)
⇒ P(E1|E) = (1/4 × 22/425) / {(1/4 × 22/425) + (1/4 × 26/425) + (1/4 × 26/425) + (1/4 × 26/425)}
⇒ P(E1|E) = 22/100 = 0.22
Hence, The required probability is 0.22.
Example 4: Suppose 15 men out of 300 men and 25 women out of 1000 are good orators. An orator is chosen at random. Find the probability that a male person is selected.
Solution:
Given,
- Total Men = 300
- Total Women = 1000
- Good Orators among Men = 15
- Good Orators among Women = 25
Total number of good orators = 15 (from men) + 25 (from women) = 40
Probability of selecting a male orator:
P(Male Orator) = Numbers of male orators / total no of orators = 15/40 = 3/8
Example 5: A man is known to speak the lies 1 out of 4 times. He throws a die and reports that it is a six. Find the probability that it Bayes’ actually a six.
Solution:
In a throw of a die, let
E1 = event of getting a six,
E2 = event of not getting a six and
E = event that the man reports that it is a six.
Then, P(E1) = 1/6, and P(E2) = (1 – 1/6) = 5/6
P(E|E1) = probability that the man reports that six occurs when six has actually occurred
⇒ P(E|E1) = probability that the man speaks the truth
⇒ P(E|E1) = 3/4
P(E|E2) = probability that the man reports that six occurs when six has not actually occurred
⇒ P(E|E2) = probability that the man does not speak the truth
⇒ P(E|E2) = (1 – 3/4) = 1/4
Probability of getting a six ,given that the man reports it to be six
P(E1|E) = P(E|E1) × P(E1)/P(E|E1) × P(E1) + P(E|E2) × P(E2) [by Bayes theorem]
⇒ P(E1|E) = (3/4 × 1/6)/{(3/4 × 1/6) + (1/4 × 5/6)}
⇒ P(E1|E) = (1/8 × 3) = 3/8
Hence the probability required is 3/8.
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Integrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface
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Fundamental Theorem of Calculus | Part 1, Part 2
Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for
11 min read
Finding Derivative with Fundamental Theorem of Calculus
Integrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
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Evaluating Definite Integrals
Integration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
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Properties of Definite Integrals
Properties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as [Tex]\int_{a}^{b}[/Tex]f(x) = F(b) − F(a) There are many properties regarding definite integral. We will discuss each property one by one with
8 min read
Definite Integrals of Piecewise Functions
Imagine a graph with a function drawn on it, it can be a straight line or a curve or anything as long as it is a function. Now, this is just one function on the graph, can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight line
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Improper Integrals
Improper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
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Riemann Sums
Riemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
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Riemann Sums in Summation Notation
Riemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
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Trapezoidal Rule
The Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form b∫a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
13 min read
Definite Integral as the Limit of a Riemann Sum
Definite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
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Antiderivative: Integration as Inverse Process of Differentiation
An antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that: d/dx[F(x)] = f(x) In other words, F(x) is a function whose derivative is f(x). Antiderivatives include a family of function
6 min read
Indefinite Integrals
Integrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
6 min read
Particular Solutions to Differential Equations
Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
7 min read
Integration by U-substitution
Finding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
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Reverse Chain Rule
Integrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
6 min read
Partial Fraction Expansion
If f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
9 min read
Trigonometric Substitution: Method, Formula and Solved Examples
Trigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
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Chapter 8: Applications of Integrals
Area under Simple Curves
We know how to calculate the areas of some standard curves like rectangles, squares, trapezium, etc. There are formulas for areas of each of these figures, but in real life, these figures are not always perfect. Sometimes it may happen that we have a figure that looks like a square but is not actual
6 min read
Area Between Two Curves: Formula, Definition and Examples
Area Between Two Curves in Calculus is one of the applications of Integration. It helps us calculate the area bounded between two or more curves using the integration. As we know Integration in calculus is defined as the continuous summation of very small units. The topic "Area Between Two Curves" h
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Area between Polar Curves
Coordinate systems allow the mathematical formulation of the position and behavior of a body in space. These systems are used almost everywhere in real life. Usually, the rectangular Cartesian coordinate system is seen, but there is another type of coordinate system which is useful for certain kinds
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Area as Definite Integral
Integrals are an integral part of calculus. They represent summation, for functions which are not as straightforward as standard functions, integrals help us to calculate the sum and their areas and give us the flexibility to work with any type of function we want to work with. The areas for the sta
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Chapter 9: Differential Equations
Differential Equations
A differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur
13 min read
Particular Solutions to Differential Equations
Indefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
7 min read
Homogeneous Differential Equations
Homogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) i
9 min read
Separable Differential Equations
Separable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y' = f(x).g(y), is called a separable differential equation. Basic
8 min read
Exact Equations and Integrating Factors
Differential Equations are used to describe a lot of physical phenomena. They help us to observe something happening in real life and put it in a mathematical form. At this level, we are mostly concerned with linear and first-order differential equations. A differential equation in “y†is linear if
10 min read
Implicit Differentiation
Implicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
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Implicit differentiation - Advanced Examples
In the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve
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Advanced Differentiation
Derivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
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Disguised Derivatives - Advanced differentiation | Class 12 Maths
The dictionary meaning of “disguise†is “unrecognizableâ€. Disguised derivative means “unrecognized derivativeâ€. In this type of problem, the definition of derivative is hidden in the form of a limit. At a glance, the problem seems to be solvable using limit properties but it is much easier to solve
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Derivative of Inverse Trigonometric Functions
Derivative of Inverse Trigonometric Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation
11 min read
Logarithmic Differentiation
Method of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma
8 min read