Probability

Why This Chapter Matters

Probability is the mathematics of uncertainty. Class 9 begins experimentally: instead of assuming what should happen, perform an experiment repeatedly and observe what actually happens.

Flip a coin 20 times and you may not get exactly 10 heads. Flip it 2,000 times and the proportion of heads is likely to become more stable. This observed proportion is experimental probability.

The chapter teaches an important scientific habit: repeat, record, compare and interpret. Probability is not about predicting every individual event with certainty; it is about describing patterns across repeated uncertain events.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Fractions
• Ratios
• Frequency tables
• Data collection
• Basic idea of chance

These are the tools this chapter reuses and deepens.

1. Random Experiments

A random experiment is a process whose exact outcome cannot be predicted with certainty before it is performed, even though possible outcomes may be known.

Examples include tossing a coin, rolling a die or drawing an object without looking. One performance of the experiment is called a trial.

2. Outcomes and Events

An outcome is the result of one trial. An event is a collection of outcomes of interest.

For a die roll, 'getting an even number' is an event involving 2, 4 and 6. In experimental probability, however, the main focus is how often the event actually occurs in repeated trials.

3. Experimental Probability

If an event E occurs m times in n trials, its experimental probability is m/n.

This is also called relative frequency. Because it is based on observed data, two groups performing the same experiment may obtain slightly different values.

4. Range from 0 to 1

An event that never occurred in the observed trials has experimental probability 0 for that data set. An event occurring every time has probability 1.

Since the event count cannot be negative or exceed the total number of trials, experimental probability always lies from 0 to 1.

5. Why More Trials Matter

With only a few trials, chance fluctuations can be large. Five tosses may produce four heads, giving 0.8.

With hundreds or thousands of trials, relative frequency usually becomes more stable. This does not make the sequence predictable; it makes the long-run proportion less sensitive to short random streaks.

6. Probability Is Not a Guarantee

If an event has probability about 0.7, that does not mean it must occur exactly 7 times in the next 10 trials.

Probability describes long-run tendency, not a promise about a short sequence. Random variation is normal and expected.

7. Connection with Statistics

Experimental probability is built from data, so probability and statistics naturally connect.

Statistics summarises what happened; experimental probability turns observed frequency into a measure of how often an event occurred. Later mathematics will introduce theoretical probability based on mathematical models rather than only on data.

Visualise the Core Idea

More trials usually make experimental probability more stable, though individual outcomes remain unpredictable.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary