GANITA MANJARI · GRADE 9 · PART I · CHAPTER 7

COMPARE — Experimental versus theoretical probability

Think first. Use AI purposefully. Verify always.

DISCOVER

Begin with your own prediction.

Must 20 tosses of a fair coin produce exactly ten heads? Would 1000 tosses guarantee exactly 500?

Write or sketch a first attempt. Use the model to test your assumptions before accepting a rule.

Understand

The theoretical probability of heads in independent fair tosses is $\frac{1}{2}$ . Observed proportions fluctuate. In long runs they tend to be closer to the model probability; no fixed trial count guarantees exact equality or steadily improving closeness.

Key Ideas

P (H) = \frac{1}{2}

A longer run need not be closer at every step. A run of heads does not require an immediate balancing run of tails.

Visualise

A longer run, not a guarantee

Demonstration seed

Local seeded simulation of independent fair coin tosses. These are generated observations, not textbook experimental data. A fixed seed reproduces the same sequence.

Trials: 0 · Heads: 0 · Tails: 0

No trials yet: observed relative frequency is not defined.

Last 0 results: None yet

Long-run relative frequency tends towards the model value; exact equality at a set number of trials is not guaranteed.

Method

Keep the event and experiment the same. Compare the observed ratio with the theoretical value. Examine several runs and sample sizes instead of selecting one convenient record.

Worked Example

Given

In 12 fair-die rolls a 3 appears 3 times (Exercise 7.2 Q6).

To Find

Experimental and theoretical probabilities of a 3

Reasoning

Experimental uses observations; theoretical uses the fair-die model.

Solution

Observed: $\frac{3}{12}$ = $\frac{1}{4}$ .
Theoretical: $\frac{1}{6}$ .
The difference is consistent with short-run variability; it does not force the next outcome.

Result

$\frac{1}{4}$ observed versus $\frac{1}{6}$ theoretical; neither implies a guaranteed next roll.

AIMS Method

A — Attempt independently
Predict and describe what is being recorded.
I — Investigate the idea
Identify the model, evidence and assumptions.
M — Make connections
Connect outcomes, events, fractions and tree paths.
S — Scrutinise your reasoning
Check completeness, likelihoods and changing conditions.

Think first. Use AI purposefully. Verify always.

Focused Practice

A student says 6000 tosses must give exactly 3000 heads. Refine the claim.

Your reasoning

Continue with EXPERIMENTAL →

Check & Feedback

Explain the model, state an assumption and identify one conclusion the information does not justify.

Explain your conclusion

Textbook connection: Ganita Manjari · Grade 9 · Part I · Chapter 7 · printed page 163