CLASS 9 MATHS · CHAPTER 7 · STUDY NOTES
What Are the Chances? Probability
Understand what are the chances? probability through short explanations, labelled models and original practice. Read one topic, explain it in your own words, then check.
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Chance describes uncertainty
Use evidence to talk about an uncertain outcome.A scale from impossible to certain
Value (number)
- Number-line marks · Only horizontal position represents the number.
- 10 · (0, 0)
- 20.25 · (0.25, 0)
- 30.5 · (0.5, 0)
- 40.75 · (0.75, 0)
- 51 · (1, 0)
0 = impossible; ½ = equally likely to occur or not occur; 1 = certain. All values use the same 0–1 scale.
Read the plotted values
| Value | Height |
|---|---|
| 0 | 0 |
| 0.25 | 0 |
| 0.5 | 0 |
| 0.75 | 0 |
| 1 | 0 |
- Probability
- A number from 0 to 1 that describes how likely an event is.
- Event
- An outcome or a group of outcomes we are interested in.
“Likely” is a prediction, not a promise. Weather information can support a prediction; a guess without evidence is weaker.
Probability 0 means impossible under the model, and 1 means certain. 0.25 = ¼ = 25%. If winning has chance ½, not winning has chance ½; not winning can include a draw.
Take a moment. What do you think?
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Question
Does a 70% forecast guarantee rain?
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Start by counting what happened
Use trial data without claiming it is an exact law.An original record of 40 spins
Count (spins)
Outcome (category: 1 blue, 2 other)
- Observed counts · An illustrative dataset, not a fairness test.
- 1Blue: 18 · (1, 18)
- 2Other: 22 · (2, 22)
Blue 18; other outcomes 22; total 40. Categories are separate, not a continuous variable.
Read the plotted values
| Outcome | Count |
|---|---|
| 1 | 18 |
| 2 | 22 |
- Trial
- One performance of an experiment.
- Experimental probability
- Observed event count ÷ total trial count, with at least one trial.
In an original example, a spinner lands on blue 18 times in 40 spins. Its observed blue frequency is 18/40 = 0.45.
Record all trials, including ones you did not expect. Another run can give a different fraction. A small dataset alone cannot prove that the spinner is fair.
Use the whole trial count
18 ÷ 40 = 0.45Do not divide blue count by other-outcome count. The denominator includes every trial.
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Question
Green occurs 12 times in 30 trials. What is its experimental probability?
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Count equally likely elementary outcomes
Check fairness before using favourable ÷ total.Two favourable faces out of six
- Theoretical probability
- For equally likely elementary outcomes, favourable count ÷ total count.
For a fair six-faced die, the event “greater than 4” contains 5 and 6. Its probability is 2/6 = ⅓.
Rain and no rain are two outcomes, but they are not automatically equally likely. Two labels alone never justify ½. A biased die also needs more information than a face count.
- 1
Describe the experiment precisely.
- 2
List the elementary outcomes.
- 3
Check the equally likely assumption.
- 4
Count the favourable outcomes and divide by the total.
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Question
What is the chance of rolling a number greater than 4 on a fair die?
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A sample must represent the group
Distinguish a useful estimate from a biased survey.A fair sample reaches across groups
- Population
- The whole group we want to learn about.
- Sample
- The smaller group actually studied.
Suppose 24 of 60 surveyed pupils prefer a fruit. The sample fraction is 0.4. It estimates the wider preference only if the sample reasonably represents that group.
Asking only pupils at one fruit stall could bias the answer. A larger biased sample is still biased; choose pupils from relevant groups fairly.
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Question
Will surveying 500 friends necessarily represent the whole school?
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A fair coin does not remember a streak
Explain what more trials can and cannot tell us.A possible running frequency
Head frequency (fraction)
Trials (count)
- Illustrative observations · The line can move towards or away from ½ locally.
- Fair-coin model · The theoretical chance remains ½.
Original illustrative counts: 2, 3, 5, 8, 18 and 33 heads at the displayed trial totals. This is not a guaranteed path.
Read the plotted values
| Trials | Head frequency |
|---|---|
| 2 | 1 |
| 4 | 0.75 |
| 8 | 0.625 |
| 16 | 0.5 |
| 32 | 0.5625 |
| 64 | 0.515625 |
| Trials | Head frequency |
|---|---|
| 0 | 0.5 |
| 80 | 0.5 |
- Gambler’s fallacy
- Thinking a past streak makes the opposite outcome due in an independent trial.
With many independent trials, relative frequency tends towards the theoretical probability. It need not move closer after every single trial.
After several heads, the next independent fair toss still has chance ½ of heads. The coin does not compensate for the past.
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Question
After eight heads, is tails now more likely than heads on a fresh fair toss?
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