1900 LearnersLast updated on June 18th, 2025
Bayes' theorem
The theorem that helps determine the conditional probability of an event based on previous events is the Bayes theorem. It helps to identify how likely an event is based on another event. In this topic, we will learn more about Bayes' theorem.
What is Bayes' Theorem
Bayes' theorem is a theorem in probability and statistics and is also known as Bayes rule and Bayes. It helps in determining the probability of event A based on the already occurred event B based on the probability of B given A, probability of A, and probability of B.
Bayes’ theorem is used to calculate the conditional probability based on the hypothesis. That is Bayes' theorem is the conditional probability of the event A, given the occurrence of another event B is equal to the product of B given A and the probability of A divided by probability B, that is
P (A|B) = P(B|A) × P(A) / P(B).
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Difference Between Conditional Probability and Bayes' theorem
Throughout the topic, we heard the words conditional probability and Bayes' theorem. Now let’s learn the difference between conditional probability and Bayes' theorem.
| Conditional Probability | Bayes' theorem |
| The probability of an event happening, given that another event has already taken place. | The probability of an event based on the previous event |
| Conditional probability is calculated using the formula, P(AB = P(A∩B)P(B) | Bayes' theorem is calculated using the formula, P(A|B) = P(B|A) × P(A) / P(B) |
| It is used to measure the outcome of an event when we have known the dependence of another event is known | It is used in inferential statistics to make decisions based on the new data |
What is the statement for Bayes’ Theorem?
To find the statement for Bayes' theorem, let the sample space be S and the set of events can be E, E, E,..., En. Where all the events have a non-zero probability of occurrence and form a part of S. According to Bayes' theorem,
P(Ei|A) = P(Ei) P(A|Ei) / ΣP(Ek) P(A|Ek), i = 1, 2, 3, … , n
What is the formula for Bayes' theorem?
For the events A and B, the formula for Bayes' theorem is
P(A|B) = P(B|A) P(A) / P(B),
where P(A) and P(B) is the probability of events A and B, P(A|B) is the probability of event A when event B happens, and P(B|A) is the probability of event B when A happens.
Derivation:
Based on conditional probability, P(A|B) = P(A ∩ B) / P(B), where, P(B) ≠ 0.
P (A ∩ B) = P (B ∩ A) = P(B|A) P(A), so
P(A|B) = P(B|A) P(A) / P(B)
Real-world applications of Bayes' theorem
To update the probability of a hypothesis, condition, or event. Let’s discuss a few real-world applications of Bayes' theorem.
To identify the spam emails in the system, we use Bayes' theorem.
- In weather forecasting, we use Bayes' theorem to improve the accuracy of weather forecasting
- To interpret DNA evidence, we use Bayes' theorem in forensic science.
- In financial forecasting, Bayes' theorem we used to assess the risk and return of the investment portfolios.
Common Mistakes and How to Avoid Them in Bayes' Theorem
Mistakes are common when students work on Bayes' theorem, now let’s learn a few common mistakes. These are a few common mistakes and ways to avoid them in Bayes' theorem.
Mistake 1
Confusing with P(A|B) with P(B|A)
The confusion with P(A|B) with P(B|A) is common among the students. So it is important to understand the concept and what is P(A|B) and P(B|A). The probability of event A based on the occurred event B that is P(A|B). Whereas, P(B|A) is the probability that event B occurred in event A.
Mistake 2
Using incorrect formula variation
Students tend to use the wrong formula that does not divide the P(B|A) P(A) with P(B). Students should understand the definition of what is Bayes' theorem, the correct formula is P(A|B) = P(B|A) P(A) / P(B).
Mistake 3
Ignoring the prior probability P(A)
The prior probability P(A) is the probability of event A occurring before the new event. So it is significant to list all the events P(A), P(B|B), and P(B) before adding the values into the formula.
Mistake 4
Unable to identify the given information
Students sometimes find it difficult to understand the question to identify the components of Bayes' theorem. So it is important for the students to break down the problem and label the variables that are P(A), P(B|A), P(B).
Mistake 5
Confusing joint and conditional probabilities
Confusing joint and conditional probabilities is common among students. Joint probability is P (A∩B) is the probability of both the events, whereas conditional probability P(A|B) is the probability of A occurring given that B has occurred.
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Solved Examples of Bayes' theorem
Problem 1
A factory produces 60% of its products from Machine A and 40% from Machine B. Machine A produces 5% defective products, while Machine B produces 10% defective products. If a randomly selected product is defective, what is the probability that it was made by Machine A?
The probability that a defective product manufactured by machine A is 42.86%
Explanation
Here,
A is the product from machine A
B is the product from machine B
D is the defective product
So, P(A) = 0.6
P(B) = 0.4
P(D|A) = 0.05
P(D|B) = 0.10
Total probability of the defective product is P(D) = P(D|A) P(A) + P(D|B) P(B)
= (0.05 × 0.6 ) + (0.10 × 0.4) = 0.03 + 0.04 = 0.07
Using Bayes' theorem to find P(A|D)
P(A|D) = P(D|A) P(A) / P(D)
= 0.05 × 0.6 / 0.07 = 0.4286
Therefore, the probability of a definitive product is 42.86%
Problem 2
A test for a certain disease is 98% accurate for people who have the disease and 95% accurate for those who don’t. If 0.5% of the population has the disease, what is the probability that a person who tested positive actually has the disease?
The probability that a person who tested positive for the disease is 9%
Explanation
Here,
D is the person has the disease
D is the person doesn’t have the disease
T is the positive test result
Given,
P(D) = 0.005
P(D) = 0.995
P(positive | D) (P(+|D) = 0.98
The false positive rate, P(+| D) = 0.05
The total probability of testing positive:
P(+) = P(+|D) × P(D) + P(+|D) × P(D)
= (0.98 × 0.005) + (0.05 × 0.995)
= 0.0049 + 0.04975 = 0.055
Using Bayes' theorem to find P(D|+),
P(D|+) = P(+|D) P(D) / P(+)
= 0.98 × 0.005 / 0.05465 = 0.0896
So, the probability that a person who tested positive is 9%
Problem 3
In a school, 30% of the students study mathematics and 70% study biology. 80% of mathematics students pass an exam, while 90% of biology students pass. If a student is randomly selected and is found to have passed, what is the probability that they studied mathematics?
The probability that a student who passed studied mathematics is 27.59%
Explanation
Here,
M is the student who studies mathematics
B is a student studies biology
P is the student passes the exam
Given,
P(M) = 0.30
P(B) = 0.70
P(P|M) = 0.80
P(P|B) = 0.90
The total probability of passing;
P(P) = P(P|M) P(M) + P(P|B) P(B)
= (0.80 × 0.30) + (0.90 × 0.70
= 0.24 + 0.63 = 0.87
Using Bayes' theorem to find P(M|P),
P(M|P) = P(P|M) P(M) / P(P)
= 0.80 × 0.30 / 0.87 = 0.2759
So, the probability that a student who passed studied mathematics is 27.59%
Problem 4
In a city, 60% of people own a car, and 40% do not. Among car owners, 70% have insurance, while only 30% of non-car owners have insurance. If a randomly chosen person has insurance, what is the probability that they own a car?
The probability that a person with insurance owns a car is 77.78%
Explanation
Here, C is the person who owns a car
N is the person who doesn't own a car
I is the person who has insurance
Given,
P(C) = 0.60
P(N) = 0.40
P(I|C) = 0.70
P(I|N) = 0.30
Finding the total probability of having insurance
P(I) = P(I|C) P(C) + P(I|N)P(N)
= (0.70 × 0.60) + (0.30 × 0.40) = 0.54
Using Bayes' theorem to find P(C|I)
P(I) = P(I|C) P(C) / P(I)
= 0.70 × 0.60 / 0.54 = 0.778
The probability that a person with insurance owns a car is 77.78%
Problem 5
A company has three suppliers: S1 (50%), S2 (30%), and S3 (20%). They supply 10%, 5%, and 2% defective items, respectively. If an item is defective, what is the probability it came from S1?
The probability that a defective item came from supplier S1 is 72.46%
Explanation
Let
S1, S2, and S3 be the events that an item is from suppliers 1, 2, 3
D is it defective
Given, P(S1) = 0.50
P(S2) = 0.30
P(S3) = 0.20
P(D|S1) = 0.10
P(D|S2) = 0.05
P(D|S3) = 0.02
The total number of defective product is;
P(D) = P(D|S1)P(S1) + P(D|S2)P(S2) + P(D|S3)P(S3)
= (0.10 × 0.50) + (0.05 × 0.30) +(0.02 × 0.20) = 0.069
Using the Bayes' theorem to find P(S1|D)
P(S1|D) = P(D|S1) P(S1) / P(D) = 0.10 × 0.50 / 0.069 = 0.05 / 0.069 = 0.7246
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FAQs on Bayes' theorem
1.What is Bayes' theorem?
2.What is the formula for Bayes' theorem?
3.What is the difference between prior and posterior probabilities?
4.Is conditional probability the same as the Bayes' theorem?
5.What are the real-world applications of Bayes' theorem?
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