2.3 Chance Models and Simulation

What Exactly Are We Repeating?

One shopper wins a gift card; the next does not. Neither result tells us what the following shopper must do. A random process produces outcomes determined by chance. An outcome is the result of one trial, and an event collects the outcomes we care about. For a five-shopper trial, “win, lose, win, lose, lose” is one outcome. “At least three winners” is an event containing several different five-shopper outcomes.

Before simulating anything, choose the unit of repetition. One digit can represent one shopper, but five digits together represent one complete five-shopper trial. Counting shoppers as though they were complete trials changes the question and its denominator.

A Model That Matches the Situation

Suppose a new store gives each shopper an independent 40% chance of winning. This is a stated model for this new example, not a claim about an observed store. Ten equally likely digits make a convenient device: 0, 1, 2 and 3 mean a win; 4 through 9 mean a loss. Generate five independent digits for one trial. Record the number of wins, then repeat. A ten-sided die labeled 0–9, random-number technology or a random-digit table can all implement the same model. With cards, replace and mix after every draw so later shoppers keep the same chance.

Here is a newly generated 20-trial demonstration. The complete digit groups are shown so the count can be checked. Each group contains five digits, and the last column counts digits from 0 through 3.

Trial	Digits	Winners	Trial	Digits	Winners
1	78063	2	11	96266	1
2	41737	2	12	40598	1
3	91276	2	13	21147	3
4	15670	2	14	38908	2
5	70396	2	15	72898	1
6	74301	3	16	50926	2
7	88345	1	17	60404	2
8	31906	3	18	54055	1
9	15698	1	19	92453	2
10	33077	3	20	93636	2

Trials 6, 8, 10 and 13 satisfy the event. The count is 4, the trial total is 20, and the estimated probability of at least three winners is $4/20=0.20$. We counted qualifying five-shopper trials, not all winning shoppers. A result of 0.20 does not change the assumed individual-shopper probability of 0.40.

The Estimate Can Change; the Model Probability Does Not

Probability describes an outcome or event’s long-run relative frequency over many independent trials. A simulation supplies empirical relative frequencies that estimate that fixed probability. Twenty repetitions can give a rough estimate, and another set of twenty may give a different one. Neither set determines the exact probability of the event.

The law of large numbers says that, for independent trials, relative frequency approaches a single value as the number of trials grows. It does not promise monotonic improvement after every trial, or a compensating win after several losses. If five shoppers have lost already, the next shopper still has the model’s 40% chance. Larger independent runs make the long-run pattern easier to see without making individual results predictable.

Keep a record with the random outcomes, the event count and the total. An estimate without its denominator hides whether it came from 20, 200 or 2,000 repetitions. When comparing runs, also check that they used the same probability model and event. More repetitions cannot rescue an encoding that assigns the wrong proportion of digits to a win.

Try It Yourself

The following original practice applies the ideas above to a fresh setting. Show the reasoning requested in each part. Answers and task-specific rubrics are on the separate solutions page.

Original generated practice for the Fall 2026 course.

A Welcome-Desk Simulation

In a new model, each visitor to a welcome desk independently has a 30% chance of receiving a voucher. One complete trial contains four visitors. Estimate the probability that at least two of the four receive vouchers. The following 20 independent four-digit groups were newly generated for this task. Each digit is equally likely to be 0–9. For every part of this task, use the prescribed encoding: digits 0, 1 and 2 mean voucher; digits 3–9 mean no voucher.

2286 5035 3180 2443 0005 3763 4428 6784 5403 4121 1689 9403 0406 8540 5970 4853 8513 0459 9934 3078

Part a

Define one complete trial, one possible outcome, and the event of interest. Use the prescribed voucher digits 0, 1 and 2 and explain why this encoding matches 30%. Describe how independence is maintained and explain why the results are random.

Your reasoning

Part b

Execute the encoding on every supplied group. Record each trial’s voucher count, the number of trials satisfying the event, and the trial total. Calculate and interpret the empirical probability estimate.

Your reasoning

Part c

Run 80 additional independent four-visitor trials using a ten-sided die, random-digit technology, or newly generated random digits. Use the same prescribed encoding: 0, 1 and 2 mean voucher. Keep a complete reproducible record of all 80 outcomes and voucher counts. Combine these with the supplied 20 trials to estimate the probability from 100 trials. State your event count and denominator.

Your reasoning

Part d

Explain what the true probability of the four-visitor event means in the long run. Two classmates’ independent 100-trial estimates are 0.31 and 0.36. Explain why they may differ, what a much larger independent run is expected to show, and why a run of losses does not force a win next.

Your reasoning

Worked answer and rubric