Comprehensive theory, key formulas, diagrams, and memory aids for Probability.
Probability is the branch of mathematics that quantifies uncertainty. It assigns numbers to the likelihood of events, so that we can reason systematically about random phenomena such as coin tosses, dice rolls, card draws, and real-world uncertainties in weather, medicine, and finance.
This chapter builds on the earlier work on sets and permutations and combinations. It defines experiments, sample spaces, and events, then develops the classical definition of probability and the axioms of probability. It introduces the algebra of events, the concepts of mutually exclusive and exhaustive events, and the axiomatic approach that makes probability a rigorous mathematical theory.
The axiomatic definition of probability is the heart of this chapter. It sets up probability as a function on events satisfying three axioms, from which all other results follow. This framework is the foundation of the theory of probability and statistics, and it prepares students for conditional probability, Bayes' theorem, and random variables studied in Class 12.
An experiment that can be repeated under identical conditions and whose outcome cannot be predicted in advance is called a random experiment. The set of all possible outcomes of a random experiment is called the sample space, denoted by S.
Examples:
Each element of the sample space is called a sample point or outcome.
An event is a subset of the sample space. If the outcome of the experiment belongs to the subset, the event is said to have occurred.
Since events are subsets, set operations apply:
Events are mutually exclusive if no two of them can occur simultaneously. They are exhaustive if their union is the entire sample space S, meaning at least one of them must occur.
If a random experiment has n equally likely outcomes, and an event A has m favourable outcomes, then the probability of A is:
P(A) = m/n = (number of favourable outcomes)/(total number of outcomes)
Example: When rolling a fair die, the probability of getting an even number is 3/6 = 1/2, since the favourable outcomes are 2, 4, 6.
In the axiomatic approach, probability is defined as a function P that assigns a real number to each event, satisfying three axioms:
These axioms capture the intuitive requirements of a probability and allow the derivation of all other properties as theorems.
From the axioms, several important theorems follow:
The last formula is the addition theorem of probability, valid for any two events.
The classical formula applies when all outcomes are equally likely, which is the case for fair coins, fair dice, and random draws from well-shuffled packs. In such cases, probability reduces to a counting problem, using the techniques of permutations and combinations.
Example: The probability of drawing a king from a well-shuffled deck of 52 cards is 4/52 = 1/13, since there are 4 kings among 52 equally likely outcomes.
| Event | Description | Probability |
|---|---|---|
| Sure event S | Must always occur | 1 |
| Impossible event phi | Can never occur | 0 |
| Event A | Favourable outcomes / total | P(A) |
| Complement A' | A does not occur | 1 - P(A) |
| Mutually exclusive A, B | Cannot occur together | P(A union B) = P(A) + P(B) |
| Experiment | Sample space | Number of outcomes |
|---|---|---|
| Toss one coin | {H, T} | 2 |
| Toss two coins | {HH, HT, TH, TT} | 4 |
| Roll one die | {1, 2, 3, 4, 5, 6} | 6 |
| Roll two dice | Ordered pairs (1,1) to (6,6) | 36 |
| Draw one card | 52 cards | 52 |
graph TD
A["Probability"] --> B["Random Experiments"]
A --> C["Events"]
A --> D["Classical Probability"]
A --> E["Axiomatic Approach"]
A --> F["Theorems"]
B --> B1["Sample space S"]
B --> B2["Outcomes / sample points"]
C --> C1["Subsets of S"]
C --> C2["Mutually exclusive"]
C --> C3["Exhaustive events"]
D --> D1["P(A) = m/n"]
D --> D2["Equally likely outcomes"]
E --> E1["P(A) >= 0"]
E --> E2["P(S) = 1"]
E --> E3["Additivity for disjoint events"]
F --> F1["P(A') = 1 - P(A)"]
F --> F2["P(A union B) = P(A) + P(B) - P(A intersection B)"]
Probability provides a rigorous framework for reasoning about uncertainty. Beginning with random experiments and sample spaces, the chapter develops events as subsets, the classical counting-based probability for equally likely outcomes, and the general axiomatic definition that makes probability a coherent mathematical theory. The addition theorem and complement rule are powerful tools that handle a wide variety of problems. The ideas established here, especially the axiomatic approach and the algebra of events, are essential for the study of conditional probability, independence, and random variables in Class 12, and they underpin statistics, science, and decision-making under uncertainty.