What this chapter is about
Probability is the branch of mathematics that measures how likely an event is to happen. In earlier classes, you explored probability through experiments and observations. In Class 10, you study the theoretical approach, where you calculate probability without actually performing the experiment, using logical reasoning about equally likely outcomes.
This chapter focuses on random experiments where all outcomes have an equal chance of occurring. You learn to identify the sample space (all possible outcomes), define events, and compute the probability of simple and compound events. Real-life situations like tossing coins, rolling dice, drawing cards, and selecting items from a group form the basis of problems.
After studying this chapter, you should be able to list all outcomes of a random experiment, identify favourable outcomes for an event, calculate probability using a formula, and understand that probability always lies between 0 and 1.
Key ideas
- A random experiment is an action where the outcome cannot be predicted with certainty, but all possible outcomes are known in advance.
- The sample space is the set of all possible outcomes of an experiment. For a single die roll, the sample space is {1, 2, 3, 4, 5, 6}.
- An event is any subset of the sample space; it consists of one or more outcomes we are interested in.
- Equally likely outcomes are outcomes that have the same chance of occurring, such as getting heads or tails in a fair coin toss.
- The probability of an event equals the number of favourable outcomes divided by the total number of outcomes, when outcomes are equally likely.
- A certain event has probability 1, and an impossible event has probability 0.
- The sum of probabilities of an event and its complement (the event not happening) is always 1.
- Probability of any event lies between 0 and 1, inclusive: 0 ≤ P(E) ≤ 1.
Formulas and facts to remember
Probability of an event E: P(E) = Number of favourable outcomes / Total number of outcomes
This gives the likelihood of event E occurring when all outcomes are equally likely.
Probability of complement: P(not E) = 1 − P(E)
If you know the chance of something happening, subtract from 1 to find the chance of it not happening.
Range of probability: 0 ≤ P(E) ≤ 1
Probability is never negative and never exceeds 1.
Certain event: P(E) = 1 when E always happens.
Impossible event: P(E) = 0 when E can never happen.
Sample space for common experiments:
- One coin: {H, T} — 2 outcomes
- Two coins: {HH, HT, TH, TT} — 4 outcomes
- One die: {1, 2, 3, 4, 5, 6} — 6 outcomes
- Two dice: 36 outcomes (each die independent)
- Standard deck of cards: 52 cards (13 each of 4 suits)
Worked examples
Example 1: Drawing a ball from a bag
A bag contains 5 red balls, 3 blue balls, and 2 green balls. One ball is drawn at random. Find the probability that it is blue.
Step 1: Total number of balls = 5 + 3 + 2 = 10. Step 2: Number of blue balls = 3. Step 3: P(blue ball) = 3/10.
The probability of drawing a blue ball is 3/10 or 0.3.
Example 2: Rolling a die for a multiple of 3
A fair die is rolled once. What is the probability of getting a multiple of 3?
Step 1: Sample space = {1, 2, 3, 4, 5, 6}. Total outcomes = 6. Step 2: Multiples of 3 in this set are 3 and 6. Favourable outcomes = 2. Step 3: P(multiple of 3) = 2/6 = 1/3.
The probability is 1/3.
Example 3: Two coins tossed together
Two fair coins are tossed simultaneously. Find the probability of getting at least one head.
Step 1: Sample space = {HH, HT, TH, TT}. Total outcomes = 4. Step 2: At least one head means one or more heads: {HH, HT, TH}. Favourable outcomes = 3. Step 3: P(at least one head) = 3/4.
Alternatively, P(no head) = P(TT) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4.
Common mistakes
Counting outcomes incorrectly when two coins are tossed by treating HT and TH as the same → Treat them as distinct; order matters, giving 4 outcomes, not 3.
Forgetting that probability cannot exceed 1 → If your answer is greater than 1, recheck your count of favourable and total outcomes.
Confusing favourable outcomes with unfavourable ones when using the complement rule → P(not E) = 1 − P(E), so subtract the event's probability, not add.
Assuming dice or coins are fair without checking the problem statement → Read carefully; some problems state biased coins or loaded dice.
Adding probabilities of overlapping events directly → When events share outcomes, use the addition rule carefully or count overlapping outcomes once.
Quick revision
- Probability = Favourable outcomes / Total outcomes (for equally likely outcomes).
- P(E) + P(not E) = 1; use this to find one if the other is known.
- Sample space for one die has 6 outcomes; for two dice, 36 outcomes.
- A standard deck has 52 cards: 4 suits, 13 cards each.
- Probability is always between 0 and 1, inclusive.
- List all outcomes carefully before counting; missed outcomes cause errors.