Experimental versus theoretical results | Year 8

01 / LEARN

Understand the idea.

Observed versus predicted

Experimental probability is the observed relative frequency: successes divided by total trials. Theoretical probability comes from a model, such as equally likely faces of a fair die. Keep the observed count distinct from what the model predicts.

Expected frequency is $n \times p$ , where $n$ is the number of trials and $p$ is the model probability. It describes an average count over repeated batches, not a requirement that each batch produce exactly that number of successes.

Variation is normal

Random variation means a fair coin need not produce exactly half heads in a finite batch. With many trials, proportions tend to stabilise around the model probability, but they need not get closer after every extra trial or batch.

A difference between theory and observation prompts checks of the model, method, recording and trial count; it does not alone prove bias. To combine unequal batches, add their successes and trial counts first. An unweighted average of their percentages can give the wrong overall frequency.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

A coin gives 40 heads in 100 tosses. Compare its relative frequency with the fair-coin model.

Worked solution

1
Calculate the observed relative frequency.
$\frac{40}{100} = 0.4$
Forty heads were recorded in one hundred tosses.
2
State the fair-coin prediction.
$P (H) = \frac{1}{2} = 0.5, 100 \times \frac{1}{2} = 50$
The model gives theoretical probability one half and an expected count of fifty heads.
3
Compare observation and theory.
$0.4 < 0.5$
This batch produced fewer heads than expected; a finite difference alone does not establish that the coin is biased.

Answer

Observed 0.4 versus theoretical 0.5.

EXAMPLE 02

Worked example 2

A model gives P(head and die 6)=1/12. There are 8 successes in 120 trials.

Worked solution

1
Calculate the model's expected count.
$120 \times \frac{1}{12} = 10$
There are 120 trials and the stated success probability is one twelfth.
2
Calculate the observed relative frequency.
$\frac{8}{120} = \frac{1}{15}$
The experimental estimate uses the eight actual successes, not the ten expected ones.
3
Compare the two counts.
$Observed = 8, expected = 10$
The expected count is a model average; random variation allows this batch to differ from it.

Answer

Expected 10; observed 8; relative frequency 1/15.

03 / YOUR TURN

Try it yourself.

Check your understanding

For P(win)=0.3, find expected wins in 200 trials.

Show answer & reasoning

60.

Expected wins equal trial count times model probability: $200 \times 0.3 = 60$ . This is the expected count, not a guaranteed result for one set of 200 trials.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

An event occurs 18 times in 30 trials. Find relative frequency.
2 marks
For $P (win) = \frac{1}{4}$ , find expected wins in 240 trials.
2 marks
A fair die gives 51 sixes in 240 rolls. State experimental and theoretical probabilities.
3 marks
There are 23 successes in 80 trials, then 37 in 120 more. Find combined relative frequency.
3 marks
Using an estimated probability 0.38, predict successes in 300 trials.
2 marks
Why are exactly 10 heads not guaranteed in 20 fair tosses?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

An event occurs 18 times in 30 trials. Find relative frequency.
2 marks
$18 \div 30 = 0.6$ .
0.6.
For $P (win) = \frac{1}{4}$ , find expected wins in 240 trials.
2 marks
$240 \times \frac{1}{4} = 60$ .
60.
A fair die gives 51 sixes in 240 rolls. State experimental and theoretical probabilities.
3 marks
Actual counts define the estimate; the fair model defines theory.
$\frac{51}{240} = \frac{17}{80}$ ; theoretical $\frac{1}{6}$ .
There are 23 successes in 80 trials, then 37 in 120 more. Find combined relative frequency.
3 marks
$\frac{23 + 37}{80 + 120} = \frac{60}{200} = 0.3$ .
0.3.
Using an estimated probability 0.38, predict successes in 300 trials.
2 marks
$300 \times 0.38 = 114$ ; it remains a prediction.
114.
Why are exactly 10 heads not guaranteed in 20 fair tosses?
2 marks
A long-run model proportion does not fix one batch’s outcome.
Many counts remain possible despite an expected count of 10.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Complementary events; two-stage sample spaces; tables, trees and Venn diagrams; probabilities of combined outcomes; experimental versus theoretical results; compound-event simulations and describing results. Formal conditional probability is Year 10.

Official curriculum source ↗

In this lesson

Understand the idea.

Observed versus predicted

Variation is normal

See the method in action.

Worked example 1

Calculate the observed relative frequency.

State the fair-coin prediction.

Compare observation and theory.

Worked example 2

Calculate the model's expected count.

Calculate the observed relative frequency.

Compare the two counts.

Try it yourself.

Check your understanding

Keep practising