Introduction to Probability
Introduction to probability and theoretical probability.
Introduction to Probability — Class 10 Mathematics, Chapter 13
Chapter 13, Probability, of Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) introduces the vocabulary used to talk about chance and uncertainty: random experiments, sample spaces, events, and the two different ways of measuring how likely something is to happen — by actually repeating an experiment and counting results, or by reasoning logically about every outcome that could occur.
None of the ideas in this introduction need any calculation beyond simple fractions, but they are the foundation every probability question in Exercise 13.1 and Exercise 13.2 is built on. Get comfortable with these definitions first, and most of the chapter's later problems become a matter of carefully listing and counting outcomes.
P(E) = (number of outcomes favourable to E) ÷ (total number of possible outcomes). Every example in this lesson — and almost every question later in the chapter — is built around this single ratio.
Random Experiments and Trials
A random experiment is one where we know every possible result in advance, but we can never predict which one of those results will actually occur on any single attempt. Tossing a coin, rolling a die, and drawing a card from a shuffled deck are all classic examples — you know exactly what could happen, just not which outcome is coming next.
Each single attempt at the experiment — one toss, one roll, one draw — is called a trial. Everything else in this chapter is really about describing and counting what can happen across these trials.
Sample Space — All Possible Outcomes
The sample space of an experiment, written as S, is simply the complete list of every outcome that experiment could produce. Nothing is left out, and nothing is repeated.
Events and Elementary Events
Out of all the outcomes in a sample space, an event (denoted E) is the set of outcomes that count as a "win" for whatever you're interested in. An event can contain one outcome, several outcomes, or even every outcome in S.
- When two coins are tossed, S = {HH, HT, TH, TT}. If the event E is "getting at least one tail," then E = {HT, TH, TT}.
- When a die is rolled, if the event A is "getting a prime number," then A = {2, 3, 5}.
- When a card is drawn from a deck, if the event B is "getting a 5," then B = {5 of Spades, 5 of Hearts, 5 of Diamonds, 5 of Clubs}.
Equally Likely Events
Two or more events are equally likely when there's no reason to expect one of them to happen more often than the other. An unbiased coin landing heads or tails, a fair die showing an even or an odd number, and a card drawn at random being red or black are all equally likely pairs.
Mutually Exclusive Events
Events are called mutually exclusive when the occurrence of one of them makes it impossible for any of the others to occur at the same time. In other words, the events share no outcomes at all.
- Tossing a coin: getting a head rules out getting a tail in that same toss — mutually exclusive.
- Rolling a die: getting an even number rules out getting an odd number — mutually exclusive.
- Rolling a die: getting an even number does not rule out getting a prime number, since 2 is both even and prime — not mutually exclusive.
- Drawing a card: getting an ace rules out getting a king in that same draw — mutually exclusive.
- Drawing a card: getting a heart does not rule out getting a king, since the king of hearts is both — not mutually exclusive.
Experimental (Empirical) Probability
One way to estimate how likely an event is, is to simply run the experiment many times and see what actually happens. Suppose a coin is tossed repeatedly, and the number of heads is counted cumulatively after every 20 tosses:
| Number of trials (n) | Number of heads (f) | f / n |
|---|---|---|
| 20 | 13 | 0.65 |
| 40 | 24 | 0.6 |
| 60 | 35 | 0.58 |
| 80 | 44 | 0.55 |
| 100 | 51 | 0.51 |
As the number of trials keeps increasing, the fraction f/n — called the relative frequency — keeps drifting closer to 0.5, which is exactly 1/2. The same pattern shows up with a die: the relative frequency of any particular face approaches 1/6 the more times the die is rolled.
Experimental probability, P(E) = (Number of trials in which E happened) ÷ (Total number of trials)
Theoretical (Classical) Probability
Instead of repeating an experiment and counting results, theoretical probability simply reasons about the sample space directly. The theoretical probability of an event E is:
P(E) = n(E) ÷ n(S) = (Number of outcomes favourable to E) ÷ (Number of all possible outcomes)
Complementary Events
For any event E, the complementary event, written Ē, is the set of every outcome that is not in E. Together, E and Ē always cover the entire sample space without overlapping.
Sure (Certain) Events and Impossible Events
Every event sits somewhere between two extremes. A sure or certain event is one that will definitely happen — one of its outcomes is guaranteed to occur in every trial. An impossible event is one that can never happen, no matter how many trials are run.
Rolling a die and getting a number ≤ 6 will always happen.
P(certain event) = 1
Rolling a single die and getting a 7 can never happen.
P(impossible event) = 0
Common Mistakes to Avoid
- Mixing up experimental and theoretical probability: Experimental probability comes from actually counting what happened across real trials; theoretical probability comes from reasoning about the sample space in advance. They usually get closer to each other as the number of trials grows, but they aren't calculated the same way.
- Under-counting a multi-step sample space: Tossing two coins gives 4 outcomes (HH, HT, TH, TT), not 2 — always list every combination, not just the individual faces.
- Assuming "two categories" means "equally likely": Prime numbers and composite numbers on a die aren't equally likely (3 chances versus 2), even though there are only two groups to choose from.
- Confusing "mutually exclusive" with "complementary": Complementary events are always mutually exclusive and together cover every outcome in S. Two mutually exclusive events don't have to cover everything — "rolling a 2" and "rolling a 5" are mutually exclusive, but they're not complementary, since neither one includes 1, 3, 4, or 6.
- Writing a probability outside the 0–1 range: If your final answer for P(E) comes out negative or greater than 1, there's an error somewhere in the count — every valid probability satisfies 0 ≤ P(E) ≤ 1.
Quick Reference — Key Terms & Formulas
| Term | Meaning / Formula |
|---|---|
| Random experiment | An experiment whose exact outcome can't be predicted in advance. |
| Sample space (S) | The complete set of all possible outcomes. |
| Event (E) | A subset of S — the outcomes that count as a "win." |
| Elementary event | An event with exactly one outcome. |
| Equally likely events | Events with no reason to favour one over another. |
| Mutually exclusive events | Events that share no outcomes — disjoint sets. |
| Experimental probability | f ÷ n (frequency of the event ÷ total trials) |
| Theoretical probability | n(E) ÷ n(S) |
| Complementary event (Ē) | All outcomes not in E; P(E) + P(Ē) = 1 |
| Sure / certain event | P(E) = 1 |
| Impossible event | P(E) = 0 |
| Range of probability | 0 ≤ P(E) ≤ 1 |
What This Lesson Prepares You For
Every definition in this introduction is put to direct use in Exercise 13.1 — Theoretical Probability, where you'll calculate P(E) for coins, dice, and playing-card problems using exactly the n(E) ÷ n(S) formula covered here. From there, Exercise 13.2 applies the same ideas to more real-world situations — bags of coloured balls, lottery tickets, and defective items in a batch — where the main challenge shifts from understanding probability to carefully counting outcomes.