Bayes Theorem

Introduction to Bayes' Theorem

Bayes' Theorem is a rule in probability that helps us determine the likelihood of an event given new information. It is named after Thomas Bayes, who explained the idea. In simple words, it is like updating our guess when we learn something new.

For example, if you want to know the possibility of rain tomorrow, you first think of the normal weather. Then you see dark clouds in the sky. With these new details, you can make a better guess. Bayes' theorem does the same in mathematics by using the formula:

P(A|B)=P(B|A)×P(A)P(B)

Table of Contents

• Bayes' Theorem Statement
• Bayes' Theorem Proof
• Bayes' Theorem Formula
• Bayes' Theorem Derivation
• Examples
• Application of Bayes' Theorem
• Practice Questions
• FAQs on Bayes' Theorem

Bayes' Theorem Statement

Suppose we have some events E1,E2,E3,...,En in an example space (a collection of all possible outcomes). These events do not overlap with each other and cover the entire sample space together. Let A be another event that happens. According to Bayes' Theorem, we can get a chance for any one event Ek. It happens when we already know that the event A has happened.

In simple words, Bayes' Theorem tells us the probability of an updated probability for a cause when we know the results.

Bayes' Theorem Proof

1. From the formula of conditional probability

P(Ek|A)=P(Ek∩A)P(A)

2. Using the multiplication rule

P(Ek∩A)=P(Ek)×P(A|Ek)

3. From the total probability theorem 

P(A)=P(E1)P(A|E1)+P(E2)P(A|E2)+...+P(En)P(A|En)

4. Putting these values together, we get the Bayes' Theorem formulas

P(Ek|A)=P(Ek)×P(A|Ek)P(E1)P(A|E1)+P(E2)P(A|E2)+...+P(En)P(A|En)

Note on Bayes' Theorem 

When we apply Bayes' Theorem, we use 3 important terms:

1. Hypotheses: Different post-bike causes 

2. Prior Probability: The chance of a cause before knowing the result.

3. Posterior Probability: The chance of a cause after knowing the result.

This is why Bayes' Theorem is called the formula of causes. It helps us update our thinking and find which cause is most likely lost once we see what has happened.

 

Bayes' Theorem Formula

If A and B are 2 events, then Bayes' Theorem can be written as:

P(A|B)=P(B|A)×P(A)P(B)

• P (A|B) means the probability of A happening when we already know B happened.

• P (B|A) means the probability of B happening when A has already happened.

• So, Bayes' theorem connects these 2 probabilities and helps us update our guess.

Bayes' Theorem Derivation

1. From the derivation of conditional probability 

P(A|B)=P(A∩B)P(B),P(B)≠0
P(B|A)=P(B∩A)P(A),P(A)≠0

2. Since P(A∩B)=P(B∩A), we can write:

P(A∩B)=P(A|B)×P(B)=P(B|A)×P(A)

3. Rearranging this gives the Bayes' theorem formula:

P(A|B)=P(B|A)×P(A)P(B),P(B)≠0

Examples

Question 1: Choosing a Coin, there are 2 coins:

• Coin I: A fair coin, 1 head, 1 tail.

• Coin II: A tricky coin, which has both sides as heads.

One coin is chosen at random and tossed. The result is Head. What is the probability that the chosen coin was Coin II? 

Solution:

• Let E base 1 = Choosing coin I, and E base 2 = choosing coin II.

• Let A = Tossing a head.

We know:

P(E1)=P(E2)=12

From coin I:

P(A|E1)=12 (1 Head out of 2 sides)

From coin II:

 P(A|E2)=1 (both sides are Heads)

Using Bayes' theorem:

P(E2|A)=P(E2)×P(A|E2)P(E1)×P(A|E1)+P(E2)×P(A|E2)
P(E2|A)=(12)×112×12+12×1=1214+12=1234=23

So, the probability that the chosen coin was Coin II is ⅔.

Question 2: Student Test Result 

A student is known to answer questions correctly 70% of the time. When he doesn't know the answer, he guesses, and the chance of being correct by guessing is 25%. Suppose the student answers a question correctly. What is the probability that he actually knew the answer?

Solution:

• Let E1 = the stent knows the answer

• Let E2  = the student's guesses.

• Lat A = the student's answer is correct.

We know:

P(E1)=0.7,P(E2)=0.3

If he knows the answer:

P(A|E1)=1

If he guesses:

P(A|E2)=0.25

Using Bayes' theorem: 

P(E1|A)=P(E1)×P(A|E1)P(E1)×P(A|E1)+P(E2)×P(A|E2)

P(E1|A)=0.7×10.7×1+0.3×0.25=0.70.7+0.075=0.70.775≈0.903

So, the probability that the student really knew the answer is about 90.3%

Application of Bayes' theorem

Bayes' theorem is useful under many real-life conditions:

• Medication: Check how the test results are correct.

• Weather report: Update the possibility of rain after seeing clouds.

• Examination: To guess whether any answer is from knowledge or a guess.

• Sport: Finding which team has a better chance to win as new results come in.

• Everyday Life: Improves guesses when new facts appear.
  
  

Practice Questions

1. Two coins
   There are two coins:Coin I is fair (head/tail). The coin II has heads on each side. A coin is chosen at random and tossed. It shows the head.
   Find: the probability that the chosen coin was coin II.

2. Two boxes of knobs
   Box A has 3 red and 2 blue. Box B has 1 red and 4 blue. You choose a box of random and draw one marble. It's red.
   Find: Probably it came from box A

3. Medical exam
   In a city, 1% of people have a disease. The test is 90% accurate for sick people (it says "positive" when they are sick) and gives 5% false positivity to healthy people. You test positive.
   Find: The possibility that you really have a disease.

4. MCQ Student
   A student knows the answer with probability 0.6. If he knows, he is right with probability 1. If he does not know, he guesses between 4 options (the possibility of being correct = 0.25). He answers correctly.
   Find: He knew the opportunity.

5. Factory machines
   The M1 machine makes 70% of items with a defective rate of 2%. The machine M2 has 30% items with a defective rate of 5%. An item is found to be defective.
   Find: The probability comes from M2.
   
   

FAQs on Bayes' Theorem

1. What is the concept of Bayes' theorem?

Bayes' Theorem is a rule in mathematics that helps us find the possibility of something happening when we already know some other related information. It combines the previous information with new information.

2. How is Bayes' theorem used in real life?

Bayes' theorem is used in many real-life situations, such as:

• Doctors use it to check how a person gets a disease after seeing the results of the test.

• E-mail apps use it to find out if there is a spam message or not.

• Weather apps use it to improve the predictions of rain or sunlight.

3. Is Bayes' theorem in the 2025 syllabus?

Yes, the Bayes theorem has been included in the class 12 maths syllabus (probability chapter) in the 2025 syllabus for several boards, such as CBSE.

4. What is Bayes' theorem in the NCERT?

In the NCERT class 12 mathematics, Bayes' theorem is explained in the probability chapter. It teaches how to calculate the probability of an event by using conditions and other known probabilities.

5. What is the correct formula for Bayes' theorem?

The formula is:

P(A|B)=P(B|A)×P(A)P(B)

Where:

• 𝑃(𝐴 | 𝐵) = The possibility of event A happening when event B has already happened.

• 𝑃( B | A ) = The possibility of event B happening when event A has already happened.

• P (A) = the possibility of event A.

• P (B) = the possibility of event B.