14.1 Event- Class 11 Mathematics

Event – Meaning, Definition, and Types

An event in probability refers to one or more outcomes of a random experiment. In mathematical terms, it is a subset of the sample space (S) — the set of all possible outcomes. Understanding events forms the backbone of probability, which carries significant weightage (around 4–6 marks) in JEE Main and Advanced examinations. Mastery of this topic helps students solve problems related to independent, dependent, and combined events effectively.

For instance, when a die is rolled, the event of getting an even number (2, 4, 6) is a subset of the sample space {1, 2, 3, 4, 5, 6}. Such identification is essential when working with probability equations or theorems like Bayes’ theorem, total probability, or conditional probability.

Occurrence of an Event

An event occurs when the actual outcome of an experiment belongs to the set representing that event. If we roll a die and get a 4, then the event “getting an even number” occurs because 4 ∈ {2, 4, 6}.

Mathematical representation:
If E ⊆ S and the outcome x ∈ E, then event E is said to have occurred.

Key concept for JEE: Identifying event occurrence is crucial when substituting correct values in probability formulas such as P(E) = n(E)/n(S).

Example:
A coin is tossed twice. Find whether the event “at least one head” occurs if the outcomes are {HH, HT, TH, TT}.
If we get HT → the event “at least one head” occurs.

Types of Events

In JEE, event-based questions test conceptual clarity and algebraic manipulation of probabilities. Events are categorized as follows:

Impossible and Sure Events

• Impossible Event: The event that cannot occur under any circumstances. Example: Getting a 7 on a standard die. Probability: P = 0.
• Sure Event: The event that will always occur. Example: Getting a number less than or equal to 6. Probability: P = 1.

JEE Tip: Understanding these helps define boundary conditions in complex probability questions.

Simple Event

A simple event contains only one possible outcome. Example: Getting a 3 when rolling a die.
JEE connection: Simple events are used in forming independent events when analyzing combined experiments.

Compound Event

A compound event involves more than one outcome. Example: Getting an even number → {2, 4, 6}.
These often appear in multi-step JEE questions involving combinations and binomial probability.

Complementary Event

For any event A, the complementary event A′ includes all outcomes not in A. Formula:
P(A) + P(A′) = 1
Example: If A = “getting a head,” then A′ = “getting a tail.”

JEE Tip: Complementary events simplify problems when the probability of one event is easier to find than the other.

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur simultaneously.
Example: A = even number, B = odd number → P(A ∩ B) = 0.
Formula: For mutually exclusive A and B,
P(A ∪ B) = P(A) + P(B)

JEE Note: Mutually exclusive and independent events are not the same. Independence implies P(A ∩ B) = P(A) × P(B), whereas mutual exclusivity implies P(A ∩ B) = 0.

Algebra of Events

The algebra of events is used to express combined probabilities and solve complex problems. These relationships are based on set theory.

Union of Events

Union of A and B (A ∪ B) represents outcomes where either A or B or both occur.
Formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Example:
A = even numbers → {2, 4, 6}, B = numbers greater than 4 → {5, 6}.
A ∪ B = {2, 4, 5, 6}.
P(A ∪ B) can be found using the above formula.

Intersection of Events

Intersection (A ∩ B) represents outcomes common to both A and B.
Formula: P(A ∩ B) = P(A) × P(B) (for independent events).
Example:
A = getting a head, B = getting a tail on two tosses → independent → P(A ∩ B) = (1/2) × (1/2) = 1/4.

Complement of an Event

Complement A′ represents all outcomes where A does not occur.
Formula: P(A′) = 1 − P(A)
Example: If P(A) = 0.25, then P(A′) = 0.75.

JEE Strategy: Complementary formulas often simplify long calculations when direct probability computation is tedious.

Marks Weightage and Sample Problems for JEE

Sample Problem 1:

A card is drawn from a standard 52-card deck. Find the probability of getting a red card or a king.

Solution:

Let A = red cards (26), B = kings (4), A ∩ B = red kings (2).
P(A ∪ B) = 26/52 + 4/52 − 2/52 = 28/52 = 7/13.

Sample Problem 2:

A coin is tossed twice. Find the probability of getting at least one head.
P(A′) = probability of no head = 1/4 → P(A) = 1 − 1/4 = 3/4.

Sample Problem 3 (Advanced JEE):

If P(A) = 1/3, P(B) = 1/2, and P(A ∩ B) = 1/6, find P(A ∪ B).

Solution:
Using formula: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 1/3 + 1/2 − 1/6 = 2/3.

Sample Problem 4 (JEE Application):

Two dice are rolled. Find the probability that the sum is 7 or 11.

Solution:
Total outcomes = 36. Favourable for 7 → 6 outcomes, for 11 → 2 outcomes.
P = (6 + 2)/36 = 8/36 = 2/9.

FAQs

Q1. What is an event in probability?

An event represents one or more possible outcomes of a random experiment. Example: Rolling a die and getting an even number.

Q2. How many types of events are important for JEE?

Six main types — impossible, sure, simple, compound, complementary, and mutually exclusive — form the foundation for higher-level JEE problems.

Q3. What is the formula for the complementary event?

P(A) + P(A′) = 1. This is especially useful when calculating “at least” or “at most” problems.

Q4. How do simple and compound events differ in JEE problems?

Simple events have one outcome, while compound events have multiple outcomes. JEE often tests compound events in dice or card-based questions.

Q5. Why is the concept of event algebra important for JEE?

It forms the basis for advanced concepts like conditional probability, independence, and Bayes’ theorem — key areas carrying higher weightage.

Q6. What is the typical JEE difficulty level for this topic?

Low to moderate. However, it directly connects to high-weightage chapters like conditional probability and random variables.

Q7. Can complementary events be mutually exclusive?

Yes, because both cannot occur together; if one happens, the other cannot.

Conclusion

The concept of events forms the starting point of probability and statistical reasoning. Students preparing for JEE should practice questions involving multiple event relationships, complementary probabilities, and intersections. The algebra of events simplifies complex problems, making it easier to calculate combined probabilities. With 4–6 marks typically allocated, consistent practice and familiarity with event-based reasoning can significantly improve performance in JEE Main and Advanced.