CBSE Class 12 Maths Chapter 13 Probability Revision Notes

Key Formulas & Concepts to Remember

• Conditional Probability: P(A|B) = P(A ∩ B) / P(B), where P(B) ≠ 0. This is the probability of A occurring, given that B has already occurred.
• Multiplication Theorem of Probability: P(A ∩ B) = P(B)P(A|B) if P(B) ≠ 0, or P(A ∩ B) = P(A)P(B|A) if P(A) ≠ 0.
• Independent Events: Two events A and B are independent if P(A ∩ B) = P(A)P(B). This implies P(A|B) = P(A) and P(B|A) = P(B).
• Theorem of Total Probability: If E₁, E₂, ..., Eₙ are n mutually exclusive and exhaustive events, and A is any event, then P(A) = P(E₁)P(A|E₁) + P(E₂)P(A|E₂) + ... + P(Eₙ)P(A|Eₙ).
• Bayes' Theorem: P(Eᵢ|A) = [P(A|Eᵢ)P(Eᵢ)] / [Σ P(A|Eⱼ)P(Eⱼ)]. Used to find the probability of a cause (Eᵢ) given an effect (A).
• Mean (Expectation) of a Random Variable: E(X) = Σ xᵢ P(xᵢ).
• Variance of a Random Variable: Var(X) = E(X²) - [E(X)]² = Σ xᵢ² P(xᵢ) - [E(X)]². Standard Deviation σ = √Var(X).
• Bernoulli Trials: Independent trials with only two outcomes (success/failure) and constant probability of success (p) for each trial.
• Binomial Distribution: P(X=k) = C(n, k) p^k (1-p)^(n-k), where 'n' is the number of trials, 'k' is the number of successes, and 'p' is the probability of success. Mean = np, Variance = np(1-p).

Key Terms & Definitions

Conditional Probability

The probability of an event occurring given that another event has already occurred. Denoted as P(A|B), the probability of A given B.

Independent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, P(A ∩ B) = P(A)P(B).

Mutually Exclusive Events

Events that cannot occur at the same time. If A and B are mutually exclusive, P(A ∩ B) = 0.

Random Variable

A function that assigns a real number to each outcome of a random experiment. It can be discrete (countable values) or continuous (values in an interval).

Probability Distribution

A table or function that lists all possible values of a random variable and their corresponding probabilities. The sum of all probabilities must be 1.

Expectation (Mean)

The weighted average of the possible values of a random variable, where the weights are the probabilities of each value. It represents the long-run average.

Variance

A measure of the spread or dispersion of a probability distribution around its mean. It quantifies how much the values typically deviate from the average.

Bernoulli Trial

A single random experiment with only two possible outcomes, typically labelled 'success' and 'failure', where the probability of success remains constant for each trial.

Binomial Distribution

A discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Bayes' Theorem

A formula used to calculate conditional probability (P(Eᵢ|A)) when the reverse conditional probabilities (P(A|Eᵢ)) and prior probabilities (P(Eᵢ)) are known. It allows updating probabilities based on new evidence.

Understanding Conditional Probability and Bayes' Theorem

Probability in Class 12 goes beyond basic calculations, introducing concepts that allow us to update our beliefs based on new information. Central to this are Conditional Probability and Bayes' Theorem.

Conditional Probability, denoted as P(A|B), refers to the likelihood of an event A occurring, given that another event B has already occurred. It effectively reduces the sample space to just those outcomes where B is true. The formula is P(A|B) = P(A ∩ B) / P(B), provided P(B) ≠ 0. For example, the probability of drawing a king given that a face card has been drawn is a conditional probability. This concept leads directly to the Multiplication Theorem of Probability, which states P(A ∩ B) = P(B)P(A|B) or P(A ∩ B) = P(A)P(B|A). This theorem is crucial for finding the probability of two events happening together.

Building upon this, we encounter the Theorem of Total Probability. Imagine a scenario where an event A can occur through several mutually exclusive and exhaustive 'causes' or preliminary events E₁, E₂, ..., Eₙ. The total probability of A occurring is the sum of the probabilities of A occurring under each of these causes, weighted by the probability of each cause itself. The formula is P(A) = Σ P(A|Eᵢ)P(Eᵢ). This theorem acts as a bridge to Bayes' Theorem.

Bayes' Theorem is perhaps the most significant application of conditional probability in this chapter. It allows us to calculate the posterior probability of a cause (Eᵢ) given that an effect (A) has already been observed. In simpler terms, it helps us determine the probability of an initial event given some new evidence. The formula is P(Eᵢ|A) = [P(A|Eᵢ)P(Eᵢ)] / [Σ P(A|Eⱼ)P(Eⱼ)]. Here, P(Eᵢ) are the prior probabilities (initial beliefs about the causes), P(A|Eᵢ) are the likelihoods (probability of effect given cause), and P(Eᵢ|A) are the posterior probabilities (updated beliefs about the causes after observing the effect). Bayes' Theorem is widely used in fields like medical diagnosis, spam filtering, and machine learning, making its understanding vital for Class 12 students.

Steps to Apply Bayes' Theorem

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Worked Examples

• {"title":"Example 1: Conditional Probability","bodyMarkdown":"A box contains 5 red and 3 blue balls. If two balls are drawn without replacement, what is the probability that the second ball drawn is red, given that the first ball drawn was red?\n\nSolution:\nLet A be the event that the first ball drawn is red.\nLet B be the event that the second ball drawn is red.\nP(A) = 5/8 (5 red balls out of 8 total)\nIf the first ball drawn was red, then there are 4 red balls left and 7 total balls.\nP(B|A) = Probability of second ball being red given first was red = 4/7.\nThus, the conditional probability is 4/7."}
• {"title":"Example 2: Bayes' Theorem","bodyMarkdown":"A factory produces bolts using three machines, A, B, and C. Machine A produces 25% of the bolts, B produces 35%, and C produces 40%. Of their output, 5%, 4%, and 2% are defective, respectively. A bolt is drawn at random and found to be defective. What is the probability that it was produced by Machine A?\n\nSolution:\nLet E₁, E₂, E₃ be the events that the bolt is produced by machines A, B, C respectively.\nLet D be the event that the bolt drawn is defective.\n\nP(E₁) = 0.25, P(E₂) = 0.35, P(E₃) = 0.40\nP(D|E₁) = 0.05, P(D|E₂) = 0.04, P(D|E₃) = 0.02\n\nWe need to find P(E₁|D).\nUsing Bayes' Theorem:\nP(E₁|D) = [P(D|E₁)P(E₁)] / [P(D|E₁)P(E₁) + P(D|E₂)P(E₂) + P(D|E₃)P(E₃)]\nP(E₁|D) = [0.05 0.25] / [ (0.05 0.25) + (0.04 0.35) + (0.02 0.40) ]\nP(E₁|D) = 0.0125 / [ 0.0125 + 0.0140 + 0.0080 ]\nP(E₁|D) = 0.0125 / 0.0345 ≈ 0.3623"}

Exam Tip: Decoding Probability Problems

A common trap in probability problems is confusing conditional probability with simple intersection, or when to apply Total Probability vs. Bayes' Theorem. Always look for keywords like "given that," "if it is known that," or "when one event has already occurred" – these strongly indicate conditional probability (P(A|B)). If the question asks for the probability of a cause given an effect (e.g., "probability that it came from machine A, given it's defective"), then Bayes' Theorem is your tool. If it's the probability of an effect that can result from multiple distinct initial events, use the Theorem of Total Probability. Clearly define your events (A, B, E₁, E₂, etc.) and write down known probabilities before plugging into formulas. This systematic approach reduces errors significantly.

Practice Questions with Solutions

• Q1: If P(A) = 0.6, P(B) = 0.3, and P(A ∩ B) = 0.18, are events A and B independent? A: Yes, because P(A ∩ B) = 0.18 and P(A)P(B) = 0.6 * 0.3 = 0.18. Since P(A ∩ B) = P(A)P(B), the events are independent.
• Q2: A random variable X has the following probability distribution: X : 0 1 2 P(X) : 0.2 0.5 0.3 Find the mean (E(X)) of X. A: E(X) = (0 0.2) + (1 0.5) + (2 * 0.3) = 0 + 0.5 + 0.6 = 1.1
• Q3: What are the conditions for a sequence of trials to be called Bernoulli trials? A: The conditions are: (i) There should be a finite number of trials. (ii) The trials must be independent. (iii) Each trial has exactly two outcomes: success or failure. (iv) The probability of success (p) remains constant for each trial.
• Q4: How does Bayes' Theorem differ from the Multiplication Theorem of Probability? A: The Multiplication Theorem calculates the probability of the intersection of two events (P(A ∩ B)). Bayes' Theorem, on the other hand, calculates the conditional probability of a 'cause' given an 'effect' (P(E|A)), effectively reversing the conditioning in P(A|E) using prior probabilities.

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