Questions
2–4 questions in board papers
Difficulty
Easy
Importance
Low-effort, high-scoring unit
Overview
Probability is the mathematical study of uncertainty, quantifying the likelihood of an event's occurrence within a defined experiment. It is a fundamental unit in the CBSE curriculum that links logical reasoning with arithmetic, often appearing in board exams as reliable, high-scoring numerical problems.
Classical Probability and Theoretical Definition
Classical probability is based on the assumption that all outcomes of an experiment are equally likely. It provides a simple ratio-based framework to measure the chance of an event happening by dividing favorable outcomes by the total possible outcomes.
- Formula: P(E) = Number of favorable outcomes / Total number of possible outcomes
- The probability of any event always lies in the range [0, 1]
- P(E) = 0 denotes an impossible event
- P(E) = 1 denotes a certain or sure event
- Total probability of all mutually exclusive and exhaustive events is always 1
Sample Space and Random Experiments
A random experiment is an action where the outcome cannot be predicted with certainty. The sample space (denoted by S) is the exhaustive set of all possible outcomes of such an experiment, serving as the denominator in probability calculations.
- Sample space of a coin toss: {H, T}
- Sample space of a single die roll: {1, 2, 3, 4, 5, 6}
- For two coins: {HH, HT, TH, TT}
- Sample space size for n coins is 2 raised to the power n
- Sample space size for n dice is 6 raised to the power n
Complementary Events
The complement of an event E, denoted as E' or not E, represents the event that E does not occur. This concept is vital for simplifying calculations where it is easier to find the probability of the negation of an event than the event itself.
- Relation: P(E) + P(not E) = 1
- P(not E) = 1 - P(E)
- Crucial for 'at least one' type problems
- Example: Probability of not getting a 6 on a die = 1 - 1/6 = 5/6
Formula Sheet
P(E) = n(E) / n(S)
P(E) + P(E') = 1
Exam Tip
Always define your sample space explicitly before calculating favorable outcomes to avoid silly errors in counting.
Common Mistakes
- Mistaking the sample space for two dice as 12 instead of 36
- Forgetting to simplify the final fraction, leading to potential marks deduction
- Confusing 'at least one' with 'exactly one' in multi-event scenarios
More Revision Notes
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