Chapter 13: Probability

In Class XI you encountered the basics: sample space, events, the axioms of probability, simple computation by counting. This chapter extends in two directions. First, conditional probability: how does new information change our probability estimates? Second, random variables and their distributions: how do we describe a numerical outcome of an experiment, and what is its expected value?

Conditional probability is the workhorse of modern reasoning under uncertainty. The conditional probability of $A$ given $B$, written $P(A \mid B)$, is the probability of $A$ in the new world where we know $B$ has occurred. The defining formula is $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$, valid when $P(B) > 0$. Reorganising gives the multiplication rule: $P(A \cap B) = P(A \mid B) P(B)$.

From these two ideas grow two of the most useful tools in elementary probability: the law of total probability and Bayes' theorem. Total probability lets us compute $P(A)$ when $A$ can occur in several mutually exclusive ways. Bayes' theorem lets us reverse a conditional probability: given $P(A \mid B)$ we can compute $P(B \mid A)$, given a few other ingredients. Bayes is the heart of statistical inference, medical diagnosis, and machine learning.

The chapter ends with random variables , numerical functions on the sample space , their probability distributions, expected value, and variance. We focus on the binomial distribution, the model for independent Bernoulli trials.

For board exams, expect Bayes' theorem problems (often dressed as medical tests or quality-control scenarios) and binomial distribution problems. For JEE the same ideas appear in cleverly stated combinatorial questions; counting and probability merge.

Prerequisites: counting principles, permutations and combinations from Class XI; basic set theory; comfort with $\binom{n}{k}$.

What's inside

1. Conditional probability , $P(A \mid B) = P(A \cap B)/P(B)$, properties.
2. Multiplication theorem , for two and more events.
3. Independent events , $P(A \cap B) = P(A) P(B)$ as a definition.
4. Total probability , partition the sample space.
5. Bayes' theorem , reversing the conditional.
6. Random variables and expectation , discrete distributions, $E[X], \mathrm{Var}(X)$.
7. Binomial distribution , $n$ independent Bernoulli trials.

Key results / Formula card

How to read this chapter

Many probability problems can be solved by careful enumeration and the multiplication rule. Sketch tree diagrams when in doubt. For Bayes' theorem, always identify clearly the prior, the likelihood, and the marginal , then plug in. For the binomial distribution, identify $n$, $p$, and the specific $k$ asked. The arithmetic is mechanical; the modelling is the harder part.