Year 7AC9M7P02

Chance Experiments and Simulations

Conduct repeated chance experiments and run simulations with a large number of trials using digital tools; compare predictions about outcomes with observed results, explaining the difference.

Theory says what should happen. An experiment shows what did. They rarely match exactly, and the reason why is the lesson.

Builds onYr 7 · Probability of Single-Step Events

Method

Expected outcomes

\text{expected} = P(E) \times \text{number of trials}

What the theory predicts.

Relative frequency

\frac{\text{times it happened}}{\text{trials}}

What actually happened.

More trials, closer match

Small numbers of trials vary wildly. This is why simulations use hundreds.

Worked example

e.g.

\text{A coin is tossed 50 times and lands heads 21 times. Compare with the theory.}

Write the theoretical probability. $P(\text{head}) = \tfrac{1}{2}$
Work out the expected number. $0{,}5 \times 50 = 25$
Work out the relative frequency from the experiment. $\frac{21}{50} = 0{,}42$
Compare the two. $0{,}42 \text{ against } 0{,}5$
Explain the difference. $\text{natural variation — 50 trials is a small number}$
Say what would reduce it. $\text{run many more trials}$

Practice

Compare what should happen with what did.

\text{Expected sixes in } 60 \text{ rolls of a die}

Answer

10

\text{A die shows a six } 15 \text{ times in } 60. \text{ Relative frequency?}

Answer

\frac{15}{60} = 0{,}25

\text{Does that prove the die is unfair?}

Answer

\text{Not on its own. Run many more trials.}

\text{Expected heads in } 200 \text{ tosses}

Answer

100

\text{Why do simulations use hundreds of trials?}

Answer

\text{Estimates settle closer to the theory as trials rise.}