CBSE Class 12 Probability & Bayes' Theorem — Complete Guide with Solved Problems

In short: Probability (Chapter 13) is one of the most scoring chapters in CBSE Class 12 Maths, carrying 6-8 marks out of 80. It covers conditional probability, Bayes' theorem, multiplication rule, independent events, random variables, and binomial distribution. The Bayes' theorem question is practically guaranteed as a 5-mark problem every year. This chapter is pattern-based — once you learn the template for Bayes' problems and binomial distribution, full marks are very achievable.

Marks Distribution

For complete marks distribution, see CBSE Class 12 Maths chapter-wise marks weightage.

Conditional Probability

The probability of event A occurring given that event B has already occurred.

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

Properties

1. 0 ≤ P(A|B) ≤ 1
2. P(S|B) = 1 (where S is the sample space)
3. P(A'|B) = 1 - P(A|B)
4. P((A ∪ B)|C) = P(A|C) + P(B|C) - P((A ∩ B)|C)

Multiplication Rule

P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)

For three events: P(A ∩ B ∩ C) = P(A) × P(B|A) × P(C|A ∩ B)

Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the other.

Condition: P(A ∩ B) = P(A) × P(B)

Also: P(A|B) = P(A) and P(B|A) = P(B)

Key facts:

• If A and B are independent, then A and B', A' and B, A' and B' are also independent
• Independent ≠ Mutually exclusive. Mutually exclusive events are actually dependent (if one occurs, the other cannot)

Bayes' Theorem

This is the most important topic in the chapter. Expect a 5-mark question on this every year.

Statement

If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events, and A is any event with P(A) ≠ 0:

P(Eᵢ|A) = P(Eᵢ) × P(A|Eᵢ) / Σ[P(Eⱼ) × P(A|Eⱼ)]

Template for Solving Bayes' Problems

1. Identify the events: E₁, E₂, ... (the "causes" or "sources")
2. Write their prior probabilities: P(E₁), P(E₂), ...
3. Identify event A: the observation or outcome
4. Write the likelihoods: P(A|E₁), P(A|E₂), ...
5. Apply the formula
6. Calculate the total probability: P(A) = Σ P(Eᵢ)P(A|Eᵢ)
7. Find the required posterior probability

Tabular Method (Recommended for Board Exams)

Then: P(Eᵢ|A) = [P(Eᵢ) × P(A|Eᵢ)] / P(A)

Board exam tip: The tabular method is the safest approach. Examiners can see your work clearly, and even if you make an arithmetic error at the end, you get marks for the correct setup.

Theorem of Total Probability

If E₁, E₂, ..., Eₙ are mutually exclusive and exhaustive events:

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

This formula is used as a stepping stone in Bayes' theorem calculations.

Random Variables and Probability Distribution

Mean (Expectation)

E(X) = μ = Σ xᵢ P(xᵢ)

Variance

Var(X) = E(X²) - [E(X)]² Where E(X²) = Σ xᵢ² P(xᵢ)

Standard Deviation

σ = √Var(X)

Binomial Distribution

When a trial has only two outcomes (success/failure) and is repeated n times independently.

Conditions for Binomial Distribution

1. Fixed number of trials (n)
2. Each trial is independent
3. Only two outcomes: success (p) or failure (q = 1-p)
4. Probability of success is the same for each trial

Formula

P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ

Where r = 0, 1, 2, ..., n

Key Results

Important relationship: Variance = npq = np(1-p). Since 0 < p < 1, variance is always less than mean (npq < np).

Solved Examples

Example 1: Conditional Probability

Q: A bag contains 3 red and 4 blue balls. Two balls are drawn one by one without replacement. Find the probability that the second ball is red given that the first ball drawn is blue.

Solution: Let A = first ball is blue, B = second ball is red. P(B|A): After drawing 1 blue ball, remaining = 3 red + 3 blue = 6 balls. P(B|A) = 3/6 = 1/2

Example 2: Bayes' Theorem (Classic Board Pattern)

Q: A factory has 3 machines A, B, and C producing 30%, 45%, and 25% of total output respectively. The defective percentages of their outputs are 2%, 3%, and 5%. A randomly selected item is found to be defective. Find the probability it was produced by machine C.

Solution: Events: E₁ (from A), E₂ (from B), E₃ (from C) A = item is defective

P(E₃|A) = 0.0125/0.032 = 125/320 = 25/64

Example 3: Bayes' Theorem (Medical Test Pattern)

Q: Suppose 0.1% of a population has a certain disease. A test detects the disease with 99% accuracy (true positive) and gives a false positive 1% of the time. A person tests positive. What is the probability they actually have the disease?

Solution: E₁ = has disease, E₂ = does not have disease A = tests positive

P(E₁|A) = 0.00099/0.01098 = 99/1098 ≈ 0.09 or about 9%

This counter-intuitive result shows how false positives dominate when the disease is rare.

Example 4: Binomial Distribution

Q: A fair coin is tossed 6 times. Find: (a) P(exactly 4 heads) (b) Mean (c) Variance.

Solution: n = 6, p = 1/2, q = 1/2

(a) P(X = 4) = ⁶C₄ × (1/2)⁴ × (1/2)² = 15 × 1/64 = 15/64

(b) Mean = np = 6 × 1/2 = 3

(c) Variance = npq = 6 × 1/2 × 1/2 = 3/2 = 1.5

Example 5: Independent Events

Q: A and B are independent events with P(A) = 0.3 and P(B) = 0.4. Find P(A ∪ B) and P(A' ∩ B').

Solution: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = P(A) + P(B) - P(A)P(B) = 0.3 + 0.4 - (0.3)(0.4) = 0.7 - 0.12 = 0.58

P(A' ∩ B') = P(A') × P(B') = (1-0.3)(1-0.4) = 0.7 × 0.6 = 0.42

Note: P(A' ∩ B') = 1 - P(A ∪ B) = 1 - 0.58 = 0.42 ✓

Board Exam Important Questions

1. Three bags contain balls as follows: Bag I: 3 red, 5 black; Bag II: 4 red, 3 black; Bag III: 5 red, 4 black. A bag is selected at random and a ball is drawn. It is found to be red. Find the probability that the ball was drawn from Bag III. (5 marks) — Classic Bayes pattern, appears almost every year
2. A couple has two children. Find the probability that both are boys if: (a) it is known that at least one is a boy (b) the older child is a boy. (3 marks)
3. If P(A) = 3/5 and P(B) = 1/5, find P(A ∩ B) if A and B are independent events. Also find P(A|B). (2 marks)
4. Five defective bulbs are accidentally mixed with 20 good ones. Three bulbs are drawn at random. Find the probability distribution of the number of defective bulbs and hence find the mean. (5 marks)
5. A die is thrown 6 times. If getting an even number is success, find the probability of getting at least 5 successes. (3 marks)
6. Two cards are drawn from a well-shuffled deck of 52 cards without replacement. Find the probability that both are kings. (2 marks)
7. An insurance company insured 3000 scooter drivers, 5000 car drivers, and 7000 truck drivers. Probability of accident is 0.04, 0.06, and 0.15 respectively. One insured driver has an accident. Find the probability they were a car driver. (5 marks)

Frequently Asked Questions

How many marks does Probability carry?

6-8 marks out of 80 in the theory paper. It is the last chapter but one of the highest-scoring because of the predictable Bayes' theorem question.

Is Bayes' theorem guaranteed in the board exam?

Virtually yes. A 5-mark Bayes' theorem problem has appeared in nearly every CBSE board paper for the past decade. The problem types rotate between factory/machine, bag/ball, and disease/test scenarios.

How do I set up a Bayes' theorem problem quickly?

Always use the tabular method: Column 1 = events (sources), Column 2 = prior probabilities, Column 3 = likelihoods, Column 4 = product. Sum the last column to get total probability. Then divide.

What is the difference between independent and mutually exclusive events?

Independent events can occur together — P(A∩B) = P(A)×P(B). Mutually exclusive events cannot occur together — P(A∩B) = 0. Two mutually exclusive events with non-zero probabilities are always dependent.

Is binomial distribution important for boards?

Yes, expect 2-3 marks. The typical question gives you n trials and asks for P(X=r), mean, or variance. Remember: mean = np, variance = npq. If the problem says "find the mean and variance," immediately use these formulas.

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