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.
2. Random Experiments and Sample Space
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:
Tossing a coin: S = {H, T}.
Tossing two coins: S = {HH, HT, TH, TT}.
Rolling a die: S = {1, 2, 3, 4, 5, 6}.
Rolling two dice: S has 36 outcomes.
Each element of the sample space is called a sample point or outcome.
3. Events
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.
A simple event is an event with a single sample point.
A compound event has more than one sample point.
The empty set phi is the impossible event.
The full sample space S is the sure event.
Algebra of Events
Since events are subsets, set operations apply:
A union B: event that A or B (or both) occurs.
A intersection B: event that both A and B occur.
A' (complement): event that A does not occur.
A - B: event that A occurs but B does not.
A intersection B = phi: A and B are mutually exclusive events.
A union B = S and A intersection B = phi: A and B are complementary events.
Mutually Exclusive and Exhaustive Events
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.
4. Classical Definition of Probability
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)
Properties
P(A) is always between 0 and 1 inclusive.
P(phi) = 0 and P(S) = 1.
P(A') = 1 - P(A).
If A and B are mutually exclusive, P(A union B) = P(A) + P(B).
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.
5. The Axiomatic Approach to Probability
In the axiomatic approach, probability is defined as a function P that assigns a real number to each event, satisfying three axioms:
Axiom 1: P(A) >= 0 for every event A.
Axiom 2: P(S) = 1, where S is the sample space.
Axiom 3: If A1, A2, A3, ... are mutually exclusive events, then P(A1 union A2 union ...) = P(A1) + P(A2) + ...
These axioms capture the intuitive requirements of a probability and allow the derivation of all other properties as theorems.
6. Theorems of Probability
From the axioms, several important theorems follow:
P(phi) = 0: the impossible event has probability 0.
P(A') = 1 - P(A): the complement rule.
P(A) <= 1 for every event A.
If A is a subset of B, then P(A) <= P(B).
P(A - B) = P(A) - P(A intersection B).
P(A union B) = P(A) + P(B) - P(A intersection B).
The last formula is the addition theorem of probability, valid for any two events.
7. Equally Likely Outcomes
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.
Quick Revision Tables
Table 1: Basic Events and Their Probabilities
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)
Table 2: Sample Spaces of Common Experiments
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
Mind Map
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)"]
Important Diagrams (SVG)
Diagram 1: Venn Diagram of Events A and B
Diagram 2: Sample Space of Rolling Two Dice
Common Mistakes
Confusing the sample space of rolling two dice with 12 outcomes. It has 36 ordered outcomes.
Adding probabilities of two events without checking whether they are mutually exclusive. Use the addition theorem when they overlap.
Forgetting that P(A') = 1 - P(A), and instead writing P(A') = P(A).
Using the classical formula when outcomes are not equally likely, such as for biased coins or dice.
Confusing mutually exclusive with exhaustive. Mutually exclusive means no overlap; exhaustive means the union is the whole sample space.
Counting favourable outcomes with the wrong method, e.g., treating ordered pairs as unordered when order matters.
Saying P(A) can exceed 1. Probabilities always lie between 0 and 1.
Applying P(A union B) = P(A) + P(B) for overlapping events, double counting the intersection.
Treating drawing with replacement and without replacement the same when counting outcomes.
Exam Tips
Always write down the sample space first, then identify the favourable outcomes for the event.
Use the complement rule P(A') = 1 - P(A) when the complement of the event is easier to count.
For dice and cards, count using ordered pairs or the combination formula, and verify with the total count.
Apply the addition theorem P(A union B) = P(A) + P(B) - P(A intersection B) whenever events can overlap.
For mutually exclusive events, the addition theorem reduces to simple addition; check this condition first.
Practise questions with coins (2^n outcomes), dice (6 and 36 outcomes), and cards (52 outcomes) to build speed.
In "at least" problems, compute the probability of the complement and subtract from 1.
Conclusion
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.