Simulating Conditional Probability

Design and conduct repeated chance experiments and simulations using digital tools to model conditional probability and interpret results.

Some probabilities are hard to calculate and easy to imitate. A simulation replaces the theory with repeated trials — and the more trials you run, the closer the estimate gets.

Builds onYr 10 · Conditional Probability

Method

Model the situation exactly

Your random device must have the same probabilities as the real thing, or you are simulating something else.

One trial is one whole scenario

A trial runs the situation from start to finish, not just one stage of it.

Discard trials the condition rules out

P(A \mid B) = \frac{\text{trials with A and B}}{\text{trials with B}}

For a conditional probability, count only the trials where the condition happened.

More trials, closer estimate

Ten trials prove nothing. Estimates settle as the count rises — this is why digital tools are used.

A simulation estimates, it does not prove

Report it as an estimate and say how many trials produced it.

Worked example

e.g.

\text{A family has two children. Given that at least one is a girl, estimate the probability that both are girls.}

Choose a device matching the real probabilities. $\text{two coins: heads = girl, tails = boy}$
Define one trial clearly. $\text{one trial} = \text{toss both coins once}$
Write down what counts as the condition, and what counts as the event. $\text{condition: at least one head} \quad \text{event: two heads}$
Run many trials and record the outcome of each. $\text{HH},\ \text{HT},\ \text{TH},\ \text{TT} \ldots$
Throw away every trial where the condition did not happen. $\text{discard all TT results}$
Divide: event trials over remaining trials. $\frac{\text{HH count}}{\text{HH} + \text{HT} + \text{TH}}$
Compare with the theory to check the model. $\text{theory} = \tfrac{1}{3} \approx 0{,}33$
Report the estimate with the number of trials that produced it. $0{,}34 \text{ from } 500 \text{ trials}$

Practice

Design the model first, then count only the trials that qualify.

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

Answer

\text{one roll of a fair six-sided die}

\text{Why are trials failing the condition discarded rather than counted as failures?}

Answer

\text{The condition says they did not happen, so they are not part of the reduced group.}

\text{A simulation of 20 trials gives } 0{,}25. \text{ Is that a good estimate?}

Answer

\text{Too few trials. Run hundreds before trusting the second figure.}

\text{You model a } 30\% \text{ chance with digits 0 to 9. Which digits mean success?}

Answer

\text{any three of them, e.g. } 0,\ 1,\ 2

\text{Your estimate is } 0{,}52 \text{ and the theory says } 0{,}33. \text{ What is the likeliest cause?}

Answer

\text{The model does not match the situation — check the device and the condition before blaming chance.}