.webp "CBSE Class 12 Probability Questions – Conditional Probability and Bayes’ Theorem")
Conditional probability and Bayes’ theorem are especially important because they connect mathematical reasoning with practical situations. Questions may involve cards, dice, defective products, medical tests, selection of students or machines producing goods.
In this guide, we will learn the important concepts through clear explanations, solved examples, exam-style questions and practice exercises. The aim is not just to find the answer but to understand each step so that you can solve unfamiliar questions confidently.
Who should read this article? Students preparing for the CBSE Class 12 Mathematics Board Examination 2027, school tests and chapter-wise revision.
- Conditional probability and its applications
- Multiplication theorem of probability
- Independent and dependent events
- Total probability theorem
- Bayes’ theorem with solved examples
- Important exam-style questions and practice problems
Table of Contents
- Probability in the CBSE Class 12 syllabus
- Basic concepts you must know
- Conditional probability
- Multiplication theorem
- Independent and dependent events
- Theorem of total probability
- Bayes’ theorem
- Important solved questions
- Common mistakes to avoid
- Practice questions
- Last-minute revision strategy
- Frequently asked questions
1. Probability in the CBSE Class 12 Mathematics Syllabus
Probability is one of the units in the CBSE Class 12 Mathematics curriculum. For the 2026–27 academic session, the Probability unit includes the following major areas:
| Topic | What you should prepare |
|---|---|
| Conditional Probability | Finding the probability of an event when another event has already occurred. |
| Multiplication Theorem | Finding the probability of simultaneous occurrence of events. |
| Independent Events | Testing whether the occurrence of one event affects another. |
| Total Probability | Finding the probability of an event through different possible cases. |
| Bayes’ Theorem | Finding the probability of a possible cause after observing an outcome. |
Always compare your preparation with the latest official syllabus and prescribed textbook. You can check the official CBSE 2026–27 curriculum and the CBSE Class 12 sample question papers.
2. Basic Probability Concepts You Must Know
Before solving conditional probability questions, revise these basic terms.
Random Experiment
An experiment whose outcome cannot be predicted with certainty before it is performed is called a random experiment.
Examples: Tossing a coin, rolling a die and drawing a card from a properly shuffled deck.
Sample Space
The set of all possible outcomes of a random experiment is called its sample space.
For a single throw of a die:
S = {1, 2, 3, 4, 5, 6}
Event
An event is a subset of the sample space.
For example, getting an even number when a die is thrown:
A = {2, 4, 6}
Complementary Event
The complement of an event A, written as A′, represents the event that A does not occur.
P(A′) = 1 − P(A)
Intersection of Events
A ∩ B means that both events A and B occur.
Union of Events
A ∪ B means that at least one of the events A or B occurs.
3. Conditional Probability
Conditional probability is the probability of an event occurring when we already know that another event has occurred.
Suppose a card is drawn from a standard deck. If we are told that the selected card is a face card, the probability that it is a king is different from the probability of selecting a king without any additional information.
This change in information is the central idea behind conditional probability.
Formula for Conditional Probability
For two events A and B, where P(B) is not zero, conditional probability is the probability of A given B.
How to understand the formula:
We restrict our attention to outcomes in B and then find the proportion of those outcomes that also belong to A.
Example: Selecting a Card
A card is drawn randomly from a standard deck of 52 cards. Given that the selected card is a face card, find the probability that it is a king.
Solution:
- There are 12 face cards: 4 kings, 4 queens and 4 jacks.
- There are 4 kings among these 12 face cards.
Required probability = 4/12 = 1/3.
Notice that the condition changes the group of possible outcomes from all 52 cards to only 12 face cards.
Example: Throwing a Die
A die is thrown once. If the outcome is known to be greater than 3, find the probability that it is an even number.
Solution:
Sample space after applying the condition = {4, 5, 6}.
Even outcomes = {4, 6}.
Therefore, required probability = 2/3.
Important Properties
- P(A|B) is defined only when P(B) is greater than zero.
- The value of conditional probability lies between 0 and 1.
- The condition must be read carefully before selecting the sample space.
4. Multiplication Theorem of Probability
The multiplication theorem is used to calculate the probability that two events occur together.
It is particularly useful in questions involving two successive selections, drawing cards without replacement, or events occurring in a particular order.
Example: Two Cards Without Replacement
Two cards are drawn one after another from a well-shuffled deck of 52 cards without replacement. Find the probability that both cards are kings.
Solution:
Probability that the first card is a king = 4/52.
After drawing one king, 51 cards remain, including 3 kings.
Probability that the second card is a king, given that the first was a king = 3/51.
Required probability = (4/52) × (3/51).
= 12/2652 = 1/221.
Exam tip: In without-replacement questions, the total number of remaining objects changes after the first selection. Do not continue using the original denominator.
5. Independent and Dependent Events
Two events are independent if the occurrence of one event does not change the probability of the other event.
For example, tossing a coin twice gives independent outcomes. The result of the first toss does not influence the result of the second toss.
Example of Independent Events
A coin is tossed twice. Find the probability of getting a head on both tosses.
Solution:
Probability of head on the first toss = 1/2.
Probability of head on the second toss = 1/2.
Required probability = 1/2 × 1/2 = 1/4.
How to Check Whether Events Are Independent
For events A and B, check whether the probability of their intersection equals the product of their individual probabilities.
If the condition is satisfied, the events are independent.
Alternatively, when P(B) is non-zero, you can check whether knowing that B has occurred leaves the probability of A unchanged.
Independent vs Dependent Events
| Independent events | Dependent events |
|---|---|
| One event does not affect the other. | One event changes the probability of the other. |
| Example: Two separate coin tosses. | Example: Two cards drawn without replacement. |
| The multiplication rule uses the product of individual probabilities. | The conditional probability of the second event must be considered. |
6. Theorem of Total Probability
The theorem of total probability helps us calculate the probability of an event when it can happen through several different, mutually exclusive cases.
Suppose E₁, E₂, ..., Eā are mutually exclusive and exhaustive events. They form a partition of the sample space. For an event A, its probability can be found by adding the probabilities of A occurring through each case.
Example: Production by Two Machines
A factory has two machines, M₁ and M₂. Machine M₁ produces 60% of the total items, while M₂ produces 40%. The probability of an item being defective is 0.02 for M₁ and 0.05 for M₂. Find the probability that a randomly selected item is defective.
Solution:
Let D represent the event that an item is defective.
- P(M₁) = 0.60
- P(M₂) = 0.40
- P(D|M₁) = 0.02
- P(D|M₂) = 0.05
Probability of a defective item from M₁ = 0.60 × 0.02 = 0.012.
Probability of a defective item from M₂ = 0.40 × 0.05 = 0.020.
Total probability = 0.012 + 0.020 = 0.032.
Therefore, the probability of selecting a defective item is 0.032 or 3.2%.
7. Bayes’ Theorem
Bayes’ theorem is used when we know that an event has occurred and want to calculate the probability of the possible cause responsible for it.
Consider a factory that uses three machines to manufacture products. If a randomly selected product is defective, Bayes’ theorem helps us calculate the probability that it came from a particular machine.
This is different from simply calculating the probability that a machine produces a defective item. The direction of the question has changed: we know the result and want to identify its likely source.
When Should You Use Bayes’ Theorem?
- When the question gives several possible sources or causes.
- When the final outcome is already known.
- When you are asked to find the probability of a particular source after observing the outcome.
- When the question involves defective products, medical tests or selection from different groups.
Worked Example: Defective Products from Three Machines
A factory has three machines A, B and C. They produce 30%, 45% and 25% of the total production, respectively. Their defective production rates are 1%, 2% and 3%, respectively. A product selected randomly is found to be defective. Find the probability that it was produced by machine C.
Step 1: Identify the events.
- A, B and C represent the machines.
- D represents the event that the selected product is defective.
Step 2: Write the given probabilities.
- P(A) = 0.30
- P(B) = 0.45
- P(C) = 0.25
- P(D|A) = 0.01
- P(D|B) = 0.02
- P(D|C) = 0.03
Step 3: Calculate the total probability of a defective product.
From A: 0.30 × 0.01 = 0.003.
From B: 0.45 × 0.02 = 0.009.
From C: 0.25 × 0.03 = 0.0075.
Total = 0.003 + 0.009 + 0.0075 = 0.0195.
Step 4: Find the probability that the defective product came from C.
Probability of a defective product from C = 0.0075.
Required probability = 0.0075 / 0.0195.
= 75/195 = 5/13.
Therefore, the required probability is 5/13.
Important observation: The probability that a product is defective and comes from C is not the same as the probability that it came from C, given that it is defective. Read the wording carefully.
8. Important Solved Questions for CBSE Class 12
Question 1: Conditional Probability Using Cards
One card is drawn from a deck of 52 cards. Given that it is a red card, find the probability that it is a queen.
Solution: There are 26 red cards and 2 red queens. Hence, the required probability is 2/26 = 1/13.
Question 2: Two Dice
Two dice are thrown together. Given that the sum is 8, find the probability that at least one die shows 3.
Solution:
Outcomes with sum 8 are (2,6), (3,5), (4,4), (5,3), (6,2).
Favourable outcomes are (3,5) and (5,3).
Required probability = 2/5.
Question 3: Independent Events
If P(A) = 1/3 and P(B) = 1/4, and A and B are independent, find P(A ∩ B).
Solution: P(A ∩ B) = (1/3) × (1/4) = 1/12.
Question 4: Drawing Balls Without Replacement
A bag contains 5 red and 3 blue balls. Two balls are drawn one after another without replacement. Find the probability that both are red.
Solution:
First red = 5/8.
Second red, given first red = 4/7.
Required probability = (5/8) × (4/7) = 5/14.
Question 5: Bayes’ Theorem with Two Suppliers
Supplier A provides 70% of a shop’s products and supplier B provides 30%. The defective rates are 2% and 5%, respectively. If a product is found defective, find the probability that it came from supplier B.
Solution:
Probability of defective product from A = 0.70 × 0.02 = 0.014.
Probability of defective product from B = 0.30 × 0.05 = 0.015.
Total defective probability = 0.014 + 0.015 = 0.029.
Required probability = 0.015/0.029 = 15/29.
Question 6: Total Probability
A student travels to school by bus or bicycle. The probability of using a bus is 0.7 and the probability of using a bicycle is 0.3. The probability of being late when using a bus is 0.1, and when using a bicycle is 0.05. Find the probability that the student is late.
Solution:
Late by bus = 0.7 × 0.1 = 0.07.
Late by bicycle = 0.3 × 0.05 = 0.015.
Total probability = 0.07 + 0.015 = 0.085.
9. Common Mistakes Students Should Avoid
Even students who know the formulas sometimes lose marks because they misread the question or use the wrong probability.
| Common mistake | How to avoid it |
|---|---|
| Ignoring the given condition | Identify the event that is already known to have occurred. |
| Using the original total after a selection without replacement | Update the number of remaining objects. |
| Assuming all events are independent | Check the independence condition before using the product of individual probabilities. |
| Confusing total probability and Bayes’ theorem | Total probability finds the overall chance of an outcome; Bayes’ theorem works backwards to find a cause. |
| Skipping event definitions | Define A, B, D or other events before substituting values. |
| Making arithmetic errors | Keep fractions or exact decimal values until the final step. |
10. Practice Questions for Self-Assessment
Try these questions without looking at the answers first. Write the event definitions, select the correct formula and show each calculation.
Section A: Basic and Conditional Probability
- A die is thrown once. Given that the result is greater than 2, find the probability that it is even.
- A card is drawn from a deck. Given that it is a black card, find the probability that it is a king.
- Two dice are thrown. Given that the sum is 9, find the probability that one die shows 4.
Section B: Multiplication and Independence
- A bag contains 4 red and 6 white balls. Two balls are drawn without replacement. Find the probability that both are white.
- If P(A) = 0.4 and P(B) = 0.5 and the events are independent, find P(A ∩ B).
- Two coins are tossed simultaneously. Find the probability of getting exactly one head.
Section C: Total Probability and Bayes’ Theorem
- Machines A and B produce 60% and 40% of a factory’s output. Their defective rates are 2% and 4%. Find the probability that a randomly selected product is defective.
- Using the information in Question 7, find the probability that a defective product came from machine B.
- Three suppliers provide 20%, 30% and 50% of a shop’s goods. Their defective rates are 1%, 2% and 3%, respectively. Find the probability that a randomly selected item is defective.
- Using the information in Question 9, find the probability that a defective item came from the third supplier.
Answer Key
- 2/3
- 1/13
- 2/5
- 1/3
- 0.2
- 1/2
- 0.028 or 2.8%
- 4/7
- 0.023 or 2.3%
- 15/23
Self-check: If you get an answer wrong, do not simply memorise the correct result. Identify whether the error occurred while defining events, selecting the formula, substituting values or simplifying the answer.
11. How to Revise Probability Before the Board Examination
Probability requires a combination of conceptual understanding and regular problem-solving. A short, focused revision routine can help you improve accuracy.
Step 1: Revise the Meaning of Each Formula
Do not begin by memorising a long list. Understand what information each rule uses and what it helps you calculate.
Step 2: Practise One Type at a Time
Begin with conditional probability. Then practise multiplication theorem, independent events, total probability and Bayes’ theorem. Move to mixed questions after understanding each type.
Step 3: Solve Questions Without Looking at Solutions
Attempt NCERT examples and exercises first. After that, use additional questions to test whether you can identify the correct method independently.
Step 4: Maintain an Error Notebook
Write down mistakes such as using the wrong condition, forgetting to update the total number of objects or confusing a cause with an observed result. Review this notebook before a test.
Step 5: Practise Under Time Limits
Once you are comfortable with the concepts, solve a small set of mixed questions within a fixed time. This improves your speed and helps you present solutions clearly.
For a broader revision routine, read our guide on effective revision techniques for better exam preparation. You can also follow the CBSE Class 12 Study Plan 2027 to organise your preparation across subjects.
12. How to Present Probability Answers in the Board Exam
A correct final answer is important, but clear mathematical presentation helps the examiner follow your reasoning.
- Start by defining the events clearly.
- Write the given probabilities separately.
- State the relevant probability rule before substituting values.
- Show the intermediate calculations.
- Simplify the final fraction or decimal carefully.
- Write the final answer in a complete sentence when appropriate.
For Bayes’ theorem questions, a small table of sources, production proportions and conditional defective rates can help you organise the information before calculating.
13. Frequently Asked Questions
1. What is conditional probability in Class 12 Mathematics?
Conditional probability is the probability of an event occurring when another event is already known to have occurred. It restricts the possible outcomes to those satisfying the given condition.
2. What is the difference between conditional probability and independent events?
Conditional probability calculates the chance of an event under a given condition. Independent events are events for which the occurrence of one does not change the probability of the other.
3. When should I use the multiplication theorem?
Use the multiplication theorem when a question asks for the probability that two or more events occur together, especially in successive selections or dependent events.
4. What is the purpose of Bayes’ theorem?
Bayes’ theorem helps calculate the probability of a possible cause or source after an outcome has been observed.
5. What is the difference between total probability and Bayes’ theorem?
Total probability calculates the overall probability of an event through different possible cases. Bayes’ theorem uses that overall probability to find the probability of a particular case after the event is observed.
6. Are conditional probability and Bayes’ theorem important for CBSE Class 12?
Both are included in the Probability unit of the CBSE Class 12 Mathematics curriculum for 2026–27. Students should practise them along with the multiplication theorem, independent events and total probability.
7. How should I prepare for Probability questions?
Understand the concepts, practise NCERT examples and exercises, solve mixed questions and review your mistakes. Do not rely only on memorising solved answers.
14. Final Words: Build Confidence in Probability
Probability is not a chapter that becomes easy through memorisation alone. The real improvement comes when you learn to interpret the information in a question and recognise the relationship between events.
Start with simple conditional probability questions, move to multiplication and independent events, and then practise total probability and Bayes’ theorem. Work through the examples in this article and attempt the practice set independently.
Remember: every probability question becomes more manageable when you define the events clearly and choose the correct method before starting the calculation.
Keep practising, check your calculations and prepare with confidence for the CBSE Class 12 Mathematics Board Examination 2027.
About the Author
Sadashiba Sahu | Choice Better Blog
Sadashiba Sahu is a teacher and education blogger who shares practical study strategies, exam preparation guidance and learning resources to help students study smarter and achieve their academic goals.
Educational note: This article is a supplementary learning resource. Students should use the latest CBSE curriculum and NCERT textbook as their primary references.
