Probability
Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). In Class 10 you work with theoretical (classical) probability — counting favourable outcomes out of equally likely total outcomes — using familiar experiments like tossing coins, rolling dice and drawing cards. It is a short, intuitive and high-scoring chapter.
Learning objectives
- Define the theoretical probability of an event.
- Use P(E) = number of favourable outcomes ÷ total number of outcomes.
- Recall that probability always lies between 0 and 1.
- Use the complement rule P(not E) = 1 − P(E).
- Solve problems involving coins, dice and playing cards.
Key concepts
Theoretical probability
When all outcomes of an experiment are equally likely, the probability of an event E is P(E) = (number of outcomes favourable to E) ÷ (total number of possible outcomes).
Range of probability
Probability always lies between 0 and 1. The probability of an impossible event is 0, and the probability of a sure (certain) event is 1.
Complement of an event
The event 'not E' (E does not happen) is the complement of E. Their probabilities add to 1, so P(not E) = 1 − P(E).
Sum of all probabilities
The sum of the probabilities of all the elementary (individual) outcomes of an experiment is 1. For example, for a coin, P(head) + P(tail) = 1/2 + 1/2 = 1.
Important formulas
Probability of an event
P(E) = favourable outcomes ÷ total outcomes
Complement rule
P(not E) = 1 − P(E)
Range
0 ≤ P(E) ≤ 1
Key definitions
- Sure (certain) event
- An event that is bound to happen; its probability is 1.
- Impossible event
- An event that cannot happen; its probability is 0.
- Complement of an event
- The event that E does not occur, written E̅ or 'not E', with P(not E) = 1 − P(E).
Solved examples
Q1. A coin is tossed once. Find the probability of getting a head.
Solution: Total outcomes = 2 (head, tail). Favourable = 1 (head). P(head) = 1/2.
Q2. A die is rolled once. Find (i) P(getting a 3) and (ii) P(getting an even number).
Solution: Total outcomes = 6. (i) Favourable for 3 = 1, so P(3) = 1/6. (ii) Even numbers are 2, 4, 6 — 3 outcomes — so P(even) = 3/6 = 1/2.
Q3. A bag has 3 red and 2 blue balls. One ball is drawn at random. Find the probability that it is red.
Solution: Total balls = 5, favourable (red) = 3. P(red) = 3/5.
Common mistakes to avoid
- Giving a probability greater than 1 or less than 0 — always check 0 ≤ P(E) ≤ 1.
- Counting the total outcomes wrongly (e.g. forgetting a card or a die face).
- Confusing 'favourable' outcomes with the total outcomes.
- Forgetting that P(E) + P(not E) = 1.
Probability — MCQ Quiz
12 questions with instant feedback. Use number keys 1–4 to answer.
The probability of a sure event is:
Practice questions
Short answer
A die is thrown once. What is the probability of getting an odd number?
Odd numbers 1, 3, 5 give 3/6 = 1/2.
If P(E) = 0.35, what is P(not E)?
P(not E) = 1 − 0.35 = 0.65.
What is the probability of getting a tail when a coin is tossed?
1/2.
Long answer
One card is drawn from a deck of 52 cards. Find the probability that it is (i) a red card, (ii) a face card.
(i) There are 26 red cards, so P(red) = 26/52 = 1/2. (ii) Face cards (J, Q, K of each suit) number 12, so P(face card) = 12/52 = 3/13.
A bag contains 5 red, 4 green and 3 blue balls. One ball is drawn at random. Find the probability it is (i) green, (ii) not blue.
Total = 12 balls. (i) P(green) = 4/12 = 1/3. (ii) Not blue means red or green = 9 balls, so P(not blue) = 9/12 = 3/4 (or 1 − 3/12).
HOTS (Higher Order Thinking)
Two coins are tossed together. Find the probability of getting at least one head.
Outcomes: HH, HT, TH, TT (4 total). 'At least one head' excludes only TT, so favourable = 3. P = 3/4.
A number is chosen at random from 1 to 25. Find the probability that it is a multiple of 5.
Multiples of 5 from 1 to 25 are 5, 10, 15, 20, 25 — that's 5 numbers. P = 5/25 = 1/5.
Quick revision
Revision notes
- P(E) = favourable ÷ total outcomes.
- 0 ≤ P(E) ≤ 1; impossible = 0, sure = 1.
- P(not E) = 1 − P(E).
- All elementary probabilities add up to 1.
Key takeaways
- Count the total outcomes carefully — that's where most errors come from.
- Use the complement rule when 'at least one' or 'not' appears.
- A probability can never exceed 1.
Frequently asked questions
What is theoretical probability?
It is the probability calculated by reasoning about equally likely outcomes, as favourable outcomes divided by total outcomes — without performing the experiment.
Can a probability be more than 1?
No. Probability always lies between 0 and 1, inclusive.
How many cards of each type are in a standard deck?
A deck has 52 cards: 26 red and 26 black, 4 suits of 13 cards each, with 4 aces, 4 kings, 4 queens and 4 jacks.