Year 9AC9M9P03

Simulating Compound Events

Design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events.

A simulation estimates what is hard to calculate. Model it exactly, run it many times, then compare with the theory.

Builds onYr 9 · Compound Events With and Without Replacement

Method

The model must match the real probabilities

Otherwise you are simulating something else.

One trial is one whole scenario

Not one stage of it.

Relative frequency estimates probability

\frac{\text{successes}}{\text{trials}}

More trials, closer to theory

This is the entire reason for using digital tools.

Report the estimate with its trial count

0.34 from 500 trials, not just 0.34.

Worked example

e.g.

\text{Estimate the chance that a family of three children includes at least one girl.}

Choose a device with matching probabilities. $\text{three coins: heads = girl}$
Define one trial clearly. $\text{toss all three once}$
Define success. $\text{at least one head}$
Run many trials and count successes. $\text{say } 437 \text{ out of } 500$
Calculate the relative frequency. $\frac{437}{500} = 0{,}874$
Compare with the theory using the complement. $1 - \left(\tfrac{1}{2}\right)^3 = 0{,}875 \quad\checkmark$

Practice

Model, define a trial, run many, then compare with theory.

\text{What device models a } \tfrac{1}{6} \text{ chance?}

Answer

\text{One roll of a fair die.}

\text{20 trials give } 0{,}45. \text{ Trustworthy?}

Answer

\text{No. Far too few trials.}

\text{Estimate } P(\text{at least one six in three rolls}).

Answer

1 - \left(\tfrac{5}{6}\right)^3 \approx 0{,}42

\text{Your estimate is } 0{,}6 \text{ and theory says } 0{,}42. \text{ Likeliest cause?}

Answer

\text{The model does not match the situation.}

\text{Why use digital tools for simulations?}

Answer

\text{They can run thousands of trials quickly.}

\text{Why report the trial count?}

Answer

\text{It shows how much the estimate can be trusted.}