Probability — The Complete Basics
A concept-first guide to Class 10 probability: the classical definition, the P(E) formula, sample spaces, the complement rule, and solved dice, coin, and card problems.
By the PadhoDost Team · 📖 7 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 The cricket toss
Before every match, the captains gather for the toss. Nobody knows if it will be heads or tails, yet everyone agrees each side has an equal chance. That simple feeling of 'equal chance' is the whole idea of probability: a number between 0 and 1 that measures how likely something is to happen.
Probability is the branch of maths that puts a number on uncertainty. An event that is impossible gets 0, an event that is certain gets 1, and everything in between (like a coin showing heads) sits somewhere between those two. In Class 10 we study the theoretical (classical) approach, where all outcomes are assumed equally likely.
The key words: experiment, outcome, event
An experiment is an action with an uncertain result (tossing a coin, rolling a die). Each possible result is an outcome. The set of all possible outcomes is the sample space. An event is a collection of one or more favourable outcomes we care about, such as 'getting an even number' when rolling a die.
| Rule | Statement | Meaning |
|---|---|---|
| Range | 0 ≤ P(E) ≤ 1 | Probability is never negative or above 1 |
| Certain event | P(E) = 1 | Sure to happen |
| Impossible event | P(E) = 0 | Cannot happen |
| Complement | P(not E) = 1 − P(E) | Sum of an event and its complement is 1 |
How to solve any probability question
- 1Identify the experiment and list the full sample space (count total outcomes).
- 2Circle the outcomes that are favourable to the event asked.
- 3Divide favourable by total to get P(E).
- 4Simplify the fraction to its lowest terms.
- 5If 'not E' is asked, use 1 − P(E) as a shortcut.
📝 Rolling a die and drawing a card
Q1: A fair die is rolled once. Find P(getting an even number).
Sample space = {1, 2, 3, 4, 5, 6}, so total outcomes = 6.
Even numbers = {2, 4, 6}, so favourable = 3.
P(even) = 3/6 = 1/2.
Q2: One card is drawn from a well-shuffled deck of 52. Find P(a king).
There are 4 kings, total = 52.
P(king) = 4/52 = 1/13.
Q3: Using the complement rule, P(not a king) = 1 − 1/13 = 12/13.
Remember these
- ✓P(E) = favourable outcomes / total outcomes, only for equally likely outcomes.
- ✓Probability always lies between 0 and 1.
- ✓P(E) + P(not E) = 1, so P(not E) = 1 − P(E).
- ✓Sum of probabilities of all elementary outcomes of an experiment = 1.
- ✓A standard deck = 52 cards, 4 suits, 12 face cards, 13 cards per suit.
⚡ Quick check
A bag contains 5 red and 3 black balls. One ball is drawn at random. What is the probability that it is NOT red?
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