Design and interpret a probability simulation

Specify the experiment before the random device

Name the outcome space, event, measured variable and repeated unit. For a single selection from labelled tokens, one selection may be one trial. If the event concerns two selections, one complete pair is one trial. The number of trials includes failures as well as successes. Keep a run, a trial and an elementary draw distinct

A complete original model

For each of two mutually independent selections, category A has probability two fifths and B has probability three fifths. A successful trial contains exactly one A. The ordered category space is AA,AB,BA,BB; these four categories are not equally likely. A trial consists of two selections, and a run repeats that full pair N times

How many equally likely ordered digit pairs are possible when each independent digit is 0..9?

Map equal random codes to unequal categories

Take a uniform integer from 0..9. Map 0,1,2,3 to A and 4,5,6,7,8,9 to B. This gives probabilities 4/10=2/5 and 6/10=3/5. For each trial obtain two independent digits. Do not assign equal probability to A and B just because there are two names

If A must have probability two fifths, how many of the ten equally likely digits must map to A?

Two independent digits are chosen from 0 to 9. Digits 0 to 3 map to A and digits 4 to 9 map to B. How many ordered pairs give exactly one A?

Two independent selections each give A with probability two fifths and B with probability three fifths. A trial succeeds when exactly one selection is A. Find its exact model probability

Write and implement a simulator, ungraded

Before running, choose a programming application or spreadsheet and name its random-number functionality. Justify the choice for this experiment and identify a reproducibility limitation. Implement this experiment in a programming language or spreadsheet using a documented uniform-integer random source. Input a positive integer N. Generate two fresh independent digits per trial; record the pair, category results and exactly-one-A indicator. Output successes/N and retain the model, random-source settings, seed if available and full trial count. Independent implementation and its report are ungraded. Retain the application/version and actual inputs and outputs; independently check the mapping and event against all 100 ordered digit pairs before interpreting random runs. Submit the choice, rationale and checks with the report

Model algorithm and invariant

Set successes=0. For trial=1,...,N: draw u and v independently and uniformly from integers 0..9; set a=(u<4) and b=(v<4); set hit to 1 when a differs from b, otherwise 0; add hit to successes. Return successes/N. After k iterations, successes is the sum of the first k event indicators and lies in 0..k. The bounded loop terminates after N trials and uses 2N elementary draws. N=0 is invalid because the estimate would divide by zero. A programming application with bounded loops and a documented random source is a defensible choice for repeated trials. Seed control, when available, supports replay; it does not prove independence or make a saved stream new evidence. Retain the actual run settings and results rather than presenting this algorithm as an executed run

Implement the same logic in a spreadsheet

A spreadsheet is a defensible alternative when inspectable trial rows and event formulas support the audit. Its recalculation behaviour is a relevant limitation. Use two columns containing uniform random integers from 0 to 9, one row per full trial. In a third column compare whether the two tests first<4 and second<4 have different truth values; encode true as 1 and false as 0. Average that indicator column over exactly N rows. Retain or freeze the generated input values before analysing or sharing the run, since recalculation may regenerate them. Record the random tool and any seed settings rather than treating a moving spreadsheet as a fixed experiment. Compare the event formula with the independently enumerated 48 successful digit pairs; a displayed proportion alone does not validate the implementation

Use a fixed trace to debug, then random runs to estimate

A deliberately supplied digit stream is useful for verifying an algorithm, but it is not evidence from random sampling. The four fixture pairs (1,8),(3,2),(9,4),(0,5) give indicators 1,0,0,1. Count at the pair level, not by successful digits. After the logic passes, run the stated random procedure separately

How many successful complete trials occur in the displayed test fixture?

A four-trial test fixture has two successful complete trials. What relative frequency does it produce?

A programmer sets the random source to the same seed at the start of every trial. What is the main problem?

Replacement changes the conditional experiment

For two draws without replacement from five labelled tokens, two A and three B, select a uniform index from the current list, remove that item, then select a uniform index from the four remaining items. Restore all five tokens before the next pair. The selections within a pair are dependent; separate trials can be independent after reset with fresh random choices

A container has two A tokens and three B tokens. Two tokens are drawn without replacement. Find the probability of AA

A container has two A tokens and three B tokens. Two tokens are drawn with replacement between draws. Find the probability of AA

Design a stateful sampler, ungraded

Implement two draws without replacement from the five labelled tokens. State exactly when an item is removed, when the list resets and which event is recorded. Compare with a with-replacement sampler. Explain why selecting two independent category labels with unchanged probabilities would model a different experiment. Independent program construction and model justification are ungraded

Model: remove by label, then restore for a new trial

At the start of each pair, use the list [A₁,A₂,B₁,B₂,B₃]. Select an index uniformly from 0..4 and remove the chosen labelled item. Select uniformly from 0..3 in the remaining list and remove that item. Record the event using both categories, then restore the full original list for the next trial. Labelling ensures each physical token has one equal-probability index. There are 5×4=20 equally likely ordered distinct-token pairs, and AA occurs in two, giving 1/10. If each selected item is replaced immediately, there are 25 equally likely ordered pairs and AA occurs in four, giving 4/25.

Check mappings and boundaries systematically

Before a large run, exhaust all ten digit inputs and check that exactly four map to A. Check boundary digits 3 and 4. Test all 100 ordered pairs against the event predicate: exactly 48 should be successes. In the removal model, test that no token label repeats within a pair and that every new trial starts with five tokens. A longer run cannot repair a wrong event, biased mapping or incorrect reset rule

Uniform decimal digits are reduced modulo 6 to simulate six equally likely categories. Is that valid?

Interpret variation without promising closer results

An estimate is the proportion of complete successful trials. Independent repetitions under the same model vary. Larger samples tend to concentrate more tightly around the model probability, but a particular larger run need not be closer. Compare absolute differences using exact fractions before rounding. Simulation can check consequences of assumptions, but cannot establish that the assumptions describe a real situation

One run has 43 successes in 100 trials. Find its exact relative frequency

The model probability is 12/25. Run A gives 5/10 and run B gives 43/100. Which observed estimate is closer?

Evaluate and report a simulation, ungraded

Submit the experiment, event, code or spreadsheet logic, reset rule, test fixtures, random-source details, seed/reproducibility information and results for several declared run sizes. Compare with the exact probability where available; retain unfavourable runs instead of choosing only a close one. Discuss model validity and simulation variation separately. The report is ungraded

Model report boundaries

For the independent two-selection model the exact event probability is 12/25, verified by all 100 digit pairs. The fixed four-trial fixture gives 1/2 and only verifies program logic. Actual random runs must be generated and recorded separately; do not invent them or call the fixture an observed random run. A saved seed and source version support replay, but repeating that same run is not a new sample. Report all predeclared runs and explain why a discrepancy may reflect finite variation, a coding error, or an unsuitable real-world model; distinguish these using tests and evidence