New NCERT Ganita Manjari · Class 9 Maths · Part I · Chapter 7: The Mathematics of Maybe: Introduction to Probability
This chapter introduces probability as a way to measure how likely an event is. It explains random experiments, the probability scale from 0 to 1, and two objective ways to measure probability: experimental and theoretical. It also covers sample spaces, events, the law of large numbers and tree diagrams.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 37–41 of the book.
Key points
- Probability measures the likelihood (chance) of an event.
- A situation is random when we know the possible outcomes but cannot predict which one will occur.
- Probability lies between 0 and 1: 0 means impossible, 0.5 means even chance and 1 means certain.
- Experimental probability uses actual data; it is the relative frequency of the event.
- Theoretical probability uses reasoning and assumes all outcomes are equally likely.
- The probabilities of all outcomes add up to 1, so P(not E) = 1 − P(E).
- Data from a sample can be used to estimate probabilities for a whole population; a larger, representative sample gives better estimates.
- Law of Large Numbers: as the number of trials increases, experimental probability gets closer to theoretical probability.
- A coin or die has no memory; each toss or roll is independent (Gambler's Fallacy is believing otherwise).
- The sample space S lists every possible outcome once; n(S) is the sample size.
- An event is a subset of the sample space.
- A tree diagram lists all outcomes of a multi-step experiment; each path from start to end is one outcome.
Formulas
| Probability scale | 0 ≤ P(E) ≤ 1 0 impossible, 0.5 even chance, 1 certain |
| Experimental probability | Number of times the event occurred ÷ Total number of trials relative frequency |
| Theoretical probability | P(E) = Number of favourable outcomes ÷ Number of possible outcomes only when outcomes are equally likely |
| Probability using sample space | P(E) = n(E)/n(S) |
| Complement | P(not E) = 1 − P(E) e.g. P(not 4 on a die) = 5/6 |
| Probability of a path in a tree diagram | product of its branch probabilities e.g. ½ × ½ = ¼ |
Key terms
- Random experiment (trial): An action you can repeat, like tossing a coin, whose result cannot be known in advance.
- Outcome: A result of the experiment.
- Sample space: The set of all possible outcomes, written in curly brackets, e.g. {H, T}.
- Event: Any single outcome or combination of outcomes; a subset of the sample space.
- Relative frequency: The number of times an event occurred divided by the total number of trials.
- Population and sample: The population is the whole group; the sample is the part we collect data from.
- Gambler's Fallacy: The mistaken belief that a result is 'due' after a run of other results.
Diagrams to learn
- Probability scale from 0 to 1
- Table of events placed on the probability scale
- Spinner with 8 equal sectors
- Bar chart of favourite fruits of 50 students
- Graph of 1000 simulated tosses of a fair coin
- Sample space table for tossing two coins
- Tree diagrams for tossing two and three coins
- P(E) = favourable ÷ possible works only when outcomes are equally likely. With 3 red and 7 blue marbles, P(red) = 3/10, not 1/2 — count marbles, not colours.
- Coins have no memory: after 6 heads in a row, P(tails on the next flip) is still 50%.
- For two coins, {2 heads, 1 head, 0 heads} are not equally likely: 1 head happens in 2 ways (HT, TH), so P(1 head) = 1/2, not 1/3.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is the difference between experimental and theoretical probability?
Experimental probability comes from actual trials or data: times the event occurred ÷ total trials. Theoretical probability uses reasoning: favourable outcomes ÷ possible outcomes, when all outcomes are equally likely.
What is the Law of Large Numbers?
As the number of trials increases, the experimental probability gets closer and closer to the theoretical probability. For a fair coin, the fraction of heads settles near 0.5.
What is the sample space when two coins are tossed?
S = {HH, HT, TH, TT}, so n(S) = 4. HT and TH are different outcomes.
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