Conditional probability
Probability · Conditional Probability\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]
Chapter Overview
Probability measures how likely an event is. In Class XI you met sample spaces and the axioms. Here you will learn how new information changes probabilities (conditional probability), how to find the probability that several events all happen (multiplication theorem, independence), how to reason backwards from an observed result to its likely cause (Bayes’ theorem), and how to describe numerical outcomes with a random variable and its mean.
Board focus: A Bayes’ theorem question (3–5 marks), a probability distribution with its mean, and a case study on conditional probability appear regularly.
Topics
Key Concepts
- \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\), \(P(B)\ne0\).
- \(P(A\cap B)=P(A)\,P(B\mid A)=P(B)\,P(A\mid B)\).
- \(A,B\) independent \(\iff P(A\cap B)=P(A)P(B)\).
- Total probability: \(P(A)=\sum P(E_i)P(A\mid E_i)\); Bayes: \(P(E_i\mid A)=\dfrac{P(E_i)P(A\mid E_i)}{\sum_j P(E_j)P(A\mid E_j)}\).
- For a random variable \(X\) with values \(x_i\) and probabilities \(p_i\): \(\sum p_i=1\), mean \(E(X)=\sum x_ip_i\).
Formulas
Multiplication theorem
Probability · Multiplication Theorem and Independent Events\[P(A\cap B)=P(A)P(B\mid A);\quad \text{independent: }P(A\cap B)=P(A)P(B)\]
Bayes’ theorem
Probability · Total Probability and Bayes’ Theorem\[P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_jP(E_j)P(A\mid E_j)}\]
Mean of a random variable
Probability · Random Variables and Probability Distributions\[E(X)=\sum x_ip_i,\qquad \sum p_i=1\]
Key Points
At least one
Probability · Multiplication Theorem and Independent EventsFor independent events, \(P(\text{at least one})=1-\prod P(A_i')\). This is usually quicker than adding cases.
Common Mistakes
Reversing the condition
Probability · Conditional Probability\(P(A\mid B)\) divides by \(P(B)\); \(P(B\mid A)\) divides by \(P(A)\). They are equal only when \(P(A)=P(B)\).
Independent vs mutually exclusive
Probability · Multiplication Theorem and Independent EventsMutually exclusive events with positive probabilities cannot be independent: \(P(A\cap B)=0\) but \(P(A)P(B)>0\).
Solved Examples
Two workshops
Probability · Total Probability and Bayes’ TheoremWorkshop X makes 60% of a firm’s chairs and workshop Y the rest. 3% of X’s chairs and 5% of Y’s chairs are faulty. A chair chosen at random is faulty. Find the probability that it came from Y.
\(P(F)=0.6(0.03)+0.4(0.05)=0.018+0.020=0.038\). \(P(Y\mid F)=\frac{0.020}{0.038}=\frac{10}{19}\).
Mean score
Probability · Random Variables and Probability DistributionsA spinner shows 1, 2 or 5 with probabilities 1/2, 1/3, 1/6. Find the mean score.
\(E(X)=\frac12+\frac23+\frac56=\frac{3+4+5}{6}=2\).
Practice & Tests
Chapter Practice
Untimed mixed questions from all topics. Answers are revealed after you submit.
Start PracticeUnit Test – Probability
A 90-minute board-style test on the whole unit “Probability”, drawn fresh from the question bank for every attempt.
Start TestProbability – Chapter Test
A board-style chapter test drawn fresh from the question bank each time: MCQs, assertion-reason, numericals, written answers you self-check against model solutions, and a case study.
Start Test