Probability

📚 Complete Practice Guide: Probability

Probability is the mathematical study of uncertainty. It provides a numerical way to describe how likely an event is to occur. Class 11 Probability practice builds on sample spaces, events and basic probability and develops important ideas such as conditional probability, the multiplication theorem, independent events, total probability and Bayes' theorem. This DigiSadhan worksheet is designed for repeated calculation and concept practice.

1. Sample Space and Events

A sample space is the set of all possible outcomes of a random experiment. An event is a collection of outcomes from the sample space. If A is an event, its probability is a number between 0 and 1. An impossible event has probability 0 and a certain event has probability 1.

0 ≤ P(A) ≤ 1    P(S)=1    P(∅)=0

2. Complement of an Event

The complement of A, written A′ or Aᶜ, contains outcomes in the sample space that are not in A. Complement questions are often the quickest route to solving “at least one” or “not occurring” problems.

P(A′)=1−P(A)

3. Addition Rule

For two events A and B, the probability of A or B includes the overlap between them. When events are mutually exclusive, the overlap is empty and the formula becomes simpler.

P(A∪B)=P(A)+P(B)−P(A∩B)

4. Conditional Probability

Conditional probability measures the chance of A when we already know that B has occurred. The condition changes the relevant sample space, so the denominator is P(B), provided P(B) is non-zero.

P(A|B)=P(A∩B)/P(B), P(B)≠0

5. Multiplication Theorem

The conditional-probability definition can be rearranged to obtain the multiplication theorem. It is useful for finding the probability that two events happen together.

P(A∩B)=P(B)P(A|B)=P(A)P(B|A)

6. Independent Events

Events A and B are independent when the occurrence of one does not change the probability of the other. In that case, the joint probability is the product of the individual probabilities.

A and B independent ⇔ P(A∩B)=P(A)P(B)

7. Total Probability

When a sample space is partitioned into mutually exclusive and exhaustive events B₁, B₂, …, Bₙ, the probability of A can be calculated by considering each route through which A can occur.

P(A)=Σ P(Bᵢ)P(A|Bᵢ)

8. Bayes' Theorem

Bayes' theorem reverses conditional probabilities. It is particularly useful when an outcome is observed and we want to determine which possible source or cause was most likely responsible for it. The denominator is commonly evaluated using the total probability theorem.

P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ) / ΣP(Bⱼ)P(A|Bⱼ)

9. Complement and “At Least One” Problems

For repeated independent trials, it is often easier to calculate the probability of no success and subtract from 1. This technique is valuable for questions involving at least one occurrence, at least one defective item or at least one success.

P(at least one)=1−P(none)

10. Common Question Types in This Tool

The worksheet includes more than seven categories: basic probability, conditional probability, multiplication theorem, independent events, total probability, Bayes' theorem, concepts/formulae, complementary probability and application-style questions. MCQ options are independently shuffled every time the worksheet is rendered.

11. Difficulty Strategy

Begin with Easy questions on direct probability and complements. Medium questions should include conditional and multiplication formulas. Hard questions should combine total probability, Bayes' theorem or multi-step independence. Mixed mode is useful for exam simulation and revision.

12. Quick Formula Revision

P(A′)=1−P(A)

P(A∪B)=P(A)+P(B)−P(A∩B)

P(A|B)=P(A∩B)/P(B)

P(A∩B)=P(B)P(A|B)

P(A)=ΣP(Bᵢ)P(A|Bᵢ)

P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ)/ΣP(Bⱼ)P(A|Bⱼ)

13. How to Use This Worksheet

Select the number of questions, mode, topic, difficulty and timer. Press Start New Test to generate a fresh set. Every question has a Check Answer button. Short-answer questions provide a textbox. After checking, correct answers turn green and wrong answers turn red. Show Answer is locked until the question has been checked. Progress, score and accuracy update immediately. When every question is checked, the final result appears. Shuffle Whole Worksheet creates a new order, while Save as PDF and Print are separate actions.

Exam tip: Identify the events first, write the relevant formula before substituting numbers, and check whether the problem is conditional, independent or based on a partition of the sample space. In Bayes' theorem, carefully match each prior probability with the correct conditional probability. Repeated randomized practice helps improve both speed and accuracy.