Chapter 14: Probability

Probability is the mathematics of uncertainty. We use it to reason about coin tosses, dice rolls, weather forecasts, insurance, quantum mechanics, and medical diagnostics. In Class XI we lay the axiomatic foundations: random experiments, sample spaces, events, and the rules that any probability assignment must obey.

A random experiment is a process whose outcome cannot be predicted with certainty , like tossing a coin. The set of all possible outcomes is the sample space $S$. A subset of $S$ is called an event. A probability is a function $P$ that assigns to each event a number in $[0, 1]$, with $P(S) = 1$ and a rule for handling disjoint events.

The chapter develops:

1. The vocabulary: experiments, outcomes, sample space, events, mutually exclusive, exhaustive.
2. The classical definition (equally likely outcomes): $P(A) = |A|/|S|$.
3. The axiomatic definition: $0 \le P(A) \le 1$, $P(S) = 1$, $P(A \cup B) = P(A) + P(B)$ for disjoint events.
4. The addition rule for any two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
5. The complement: $P(A') = 1 - P(A)$.

We do not yet do conditional probability or independence , those wait for Class XII.

For all real-world applications (insurance, medicine, machine learning, quantum physics) probability is the basic language. The axioms you meet here serve from now on.

What's inside

1. Random experiments and sample spaces , outcomes, sample space examples.
2. Events: types and combinations , sure, impossible, simple, compound; union, intersection, complement.
3. Mutually exclusive and exhaustive events.
4. Classical probability and equally likely outcomes.
5. Axiomatic probability , formal definition.
6. Addition rule and complement; applications.

Key results / Formula card

How to read this chapter

For every probability problem: (1) describe the sample space carefully, (2) identify the event as a subset, (3) count outcomes if equally likely, or use the rules otherwise. Many mistakes come from mis-specifying the sample space , slow down on that step.