Probability for Class 10: Concepts, Formula & Examples
Probability is the branch of mathematics that puts a number on uncertainty — how likely something is to happen. It is one of the friendliest chapters in Class 10 because most questions rest on a single formula. Master that formula, learn to count outcomes carefully, and you can answer almost every board question on dice, cards and coins. This guide explains the core ideas and works through plenty of examples.
What does probability mean?
When you toss a coin you cannot say for certain whether it will land heads or tails, but you can say that each result is equally likely. Probability measures this chance on a scale from 0 to 1. A probability of 0 means the event is impossible, and a probability of 1 means it is certain. Everything else falls in between.
Before computing anything, you need two terms. An outcome is a single possible result of an experiment. The sample space is the set of all possible outcomes. An event is the particular outcome (or group of outcomes) you are interested in.
The basic probability formula
For an experiment in which all outcomes are equally likely, the probability of an event E is:
P(E) = Number of favourable outcomes ÷ Total number of outcomes
Because the favourable outcomes can never exceed the total, the answer always lies between 0 and 1. A handy check: if your answer is greater than 1 or negative, you have made a counting error.
Worked examples
Example 1: Tossing a coin
What is the probability of getting a head when a fair coin is tossed once?
The sample space is {Head, Tail}, so the total number of outcomes is 2. The favourable outcome (a head) is just 1. Therefore P(Head) = 1/2 = 0.5.
Example 2: Rolling a die
A fair six-sided die is rolled. Find the probability of getting (a) the number 4, and (b) an even number.
The sample space is {1, 2, 3, 4, 5, 6}, so the total is 6. For (a), only one face shows 4, so P(4) = 1/6. For (b), the even numbers are 2, 4 and 6 — three favourable outcomes — so P(even) = 3/6 = 1/2.
Example 3: Drawing a card
A card is drawn at random from a well-shuffled standard deck of 52 cards. Find the probability that it is (a) a king, and (b) a red card.
A deck has 52 cards in total. There are 4 kings, so P(king) = 4/52 = 1/13. There are 26 red cards (hearts and diamonds), so P(red) = 26/52 = 1/2.
Useful facts about a deck of cards
Card questions are common, so keep this breakdown ready:
| Category | Count |
|---|---|
| Total cards | 52 |
| Suits (hearts, diamonds, clubs, spades) | 4 of 13 each |
| Red cards (hearts + diamonds) | 26 |
| Black cards (clubs + spades) | 26 |
| Face cards (J, Q, K of each suit) | 12 |
| Aces | 4 |
Complementary events
The complement of an event E is the event that E does not happen, written as E′ or "not E". Since one of the two must occur, their probabilities always add up to 1:
P(E) + P(not E) = 1
This rearranges to P(not E) = 1 − P(E), which often saves work. For instance, the probability of not rolling a 4 on a die is 1 − 1/6 = 5/6, far quicker than counting the five remaining faces.
Common mistakes to avoid
- Miscounting the sample space — for two coins it is {HH, HT, TH, TT}, giving 4 outcomes, not 3.
- Forgetting that probability can never be more than 1 or less than 0.
- Treating unequal outcomes as equally likely; the basic formula assumes fairness.
Want to test yourself? Try the LearnIQ practice quiz for more mathematics questions, or browse all study guides.