Introduction
Probability is one of the most intriguing and widely applied branches of mathematics. It deals with the measurement of uncertainty, helping us predict how likely an event is to occur. This chapter, based on NCERT Class 11 Mathematics, lays the foundation for advanced topics like random variables, probability distributions, and statistics, which are frequently tested in JEE Main, JEE Advanced, and other competitive exams. Students typically find 8–10 marks worth of direct and application-based problems from probability across JEE papers.
The concept of probability originated from gambling and games of chance but now finds extensive applications in science, engineering, artificial intelligence, and finance. The chapter covers classical, empirical, and axiomatic approaches to probability along with the algebra of events and conditional probability — all of which are crucial for mastering higher-level problems.
Random Experiments and Sample Space
A random experiment is an action or process that produces an outcome that cannot be predicted with certainty, even if repeated under identical conditions.
Examples: Tossing a coin, rolling a die, drawing a card from a deck.
Each possible result of such an experiment is called an outcome, and the set of all possible outcomes is known as the sample space (S).
- Coin toss: S = {H, T}
- Die roll: S = {1, 2, 3, 4, 5, 6}
- Two coins: S = {HH, HT, TH, TT}
JEE Tip: Clearly identifying the sample space is the first step to solving any probability problem correctly.
Events and Types of Events
An event is a subset of the sample space that represents one or more outcomes. Understanding event relationships and their algebra helps in simplifying probability calculations.
Types of Events
- Simple Event: Contains only one outcome. Example: getting a 3 on a die.
- Compound Event: Contains more than one outcome. Example: getting an even number → {2, 4, 6}.
- Sure Event: Always occurs. Example: getting a number less than 7 when rolling a die.
- Impossible Event: Can never occur. Example: getting a 7 on a die.
- Complementary Events: Events that together make up the entire sample space. P(A) + P(A′) = 1.
- Mutually Exclusive Events: Cannot occur simultaneously. Example: getting an even and odd number together.
Definitions of Probability
Probability can be defined in three major ways, each with distinct applications and limitations:
1. Classical (Theoretical) Definition
If an experiment has n equally likely outcomes, and m of them correspond to event A, then:
P(A) = m/n
Example: When a die is rolled, the probability of getting an even number = 3/6 = 1/2.
This approach assumes all outcomes are equally likely.
2. Empirical (Experimental) Probability
When outcomes are observed experimentally:
P(E) = (Number of favourable outcomes) / (Total number of trials)
Used when theoretical probability is difficult to compute, such as weather forecasting or reliability studies.
3. Axiomatic (Modern) Definition
Proposed by Kolmogorov, it is based on three axioms:
- Non-negativity: P(A) ≥ 0
- Certainty: P(S) = 1
- Additivity: If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B)
JEE Focus: Axiomatic probability forms the theoretical base for advanced topics like conditional probability and independence of events.
Algebra of Events
The algebra of events helps simplify relationships between multiple events using set operations.
- Union (A ∪ B): Either A or B or both occur.
Formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) - Intersection (A ∩ B): Both A and B occur simultaneously.
- Complement (A′): Event A does not occur. P(A′) = 1 − P(A)
Example: For A = “even numbers” and B = “numbers greater than 4,”
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
JEE Application: Problems involving multiple conditions, such as “at least one,” “exactly one,” or “none,” often require algebra of events.
Conditional Probability
Conditional probability measures the likelihood of an event occurring given that another event has already occurred.
P(A | B) = P(A ∩ B) / P(B)
Example: If 60% of students like Maths (B), and 30% like both Maths and Physics (A ∩ B), then
P(A | B) = (0.3) / (0.6) = 0.5.
Real-World Uses: Used in medical testing, reliability prediction, and AI probability models.
JEE Weightage: 3–4 marks — often part of multi-step reasoning problems.
Independence of Events
Two events A and B are independent if the occurrence of one does not affect the other.
[ P(A ∩ B) = P(A) × P(B) ]
Example: Tossing two coins. The result of one toss does not affect the other.
If events are not independent, they are dependent.
Common JEE Question: Distinguish between mutually exclusive and independent events.
Bayes’ Theorem
A crucial theorem that connects conditional probabilities and allows calculation of reverse probabilities.
P(Ai | B) = [P(Ai) × P(B | Ai)] / [P(A1)P(B | A1) + P(A2)P(B | A2) + … + P(An)P(B | An)]
Example (JEE Type): A factory has three machines A₁, A₂, and A₃ producing 25%, 35%, and 40% of total items. The defect rates are 5%, 4%, and 2% respectively. Find the probability that a randomly selected defective item came from machine A₁.
JEE Importance: Bayes’ Theorem questions are highly conceptual and often appear in advanced probability sections.
Random Variables and Probability Distribution (Introductory Concept)
A random variable is a numerical representation of the outcomes of a random experiment.
- Discrete Random Variable: Takes specific values (e.g., number of heads when tossing 3 coins).
- Continuous Random Variable: Takes any value within an interval.
A probability distribution assigns probabilities to each possible value of a random variable.
Example: Tossing a coin thrice:
X = number of heads, P(X) = {0, 1, 2, 3}.
Corresponding probabilities: P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8.
Theoretical Results in Probability
Some essential theorems and relationships frequently tested in JEE:
- P(A′) = 1 − P(A)
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
- For mutually exclusive events, P(A ∪ B) = P(A) + P(B)
These theorems simplify multi-event calculations and reduce conceptual errors.
Important JEE-Level Problems
Problem 1:
A card is drawn from a standard deck. Find the probability that it is a king or a heart.
- P(A) = 4/52, P(B) = 13/52, P(A ∩ B) = 1/52
P(A ∪ B) = 4/52 + 13/52 − 1/52 = 16/52 = 4/13.
Problem 2:
A box contains 3 red and 2 blue balls. Two balls are drawn without replacement. Find the probability that both are red.
- P(1st red) = 3/5, P(2nd red) = 2/4
P(both red) = 3/5 × 2/4 = 3/10.
Problem 3:
If two dice are rolled, find the probability that the sum is a prime number.
Possible sums = 2, 3, 5, 7, 11 → 15 favourable outcomes.
P = 15/36 = 5/12.
Problem 4 (Advanced JEE):
A bag contains 5 white, 7 black, and 3 red balls. Three balls are drawn at random. Find the probability that they are of different colours.
P = (5/15) × (7/14) × (3/13) = 105/2730 = 7/182
FAQs
Q1. What is the importance of probability in JEE exams?
Probability forms part of Unit 11 in the JEE syllabus and connects directly with statistics and permutations & combinations, contributing 8–10 marks.
Q2. How many types of probability are there?
Three — classical, empirical, and axiomatic — each with different methodologies and use cases.
Q3. What is the difference between independent and mutually exclusive events?
Independent events do not influence each other, while mutually exclusive events cannot happen together.
Q4. What is conditional probability used for?
To determine the probability of an event given that another has already occurred. It is crucial for Bayes’ Theorem.
Q5. What are common mistakes students make in JEE probability problems?
Ignoring dependence between events, misunderstanding complementary cases, and neglecting set relationships.
Conclusion
Probability is both conceptual and application-oriented, bridging logic, combinatorics, and algebra. Chapter 14 introduces students to random experiments, events, and three key probability approaches. It builds the foundation for advanced topics tested in JEE and other competitive exams. Regular practice of conditional probability, Bayes’ theorem, and independence problems ensures a strong command of this topic.
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