Design and evaluate a simulation for a compound event

Learning goal: Translate a chance model into a random-digit simulation, define one trial correctly, generate and record repeated trials and compare an observed frequency with the model. Distinguish at least four attempts from exactly four attempts.

Suppose a hypothetical independent attempt has a 40% chance of finding a special token. Each attempt follows the same probability, and earlier results do not change the next attempt's chance. We ask: “What is the probability that it takes at least four attempts to find the first special token?” This is a waiting-time question whose event must be translated carefully before simulating.

At least four attempts means the first three attempts fail. The fourth may succeed, or the first success may come later. Therefore a trial can check only the first three outcomes to decide whether the event occurs. We do not need to keep drawing until a success just to answer this particular question. Exactly four attempts would add the condition that the fourth is a success, a different event.

Use equally likely digits zero through nine. Assign 0, 1, 2 and 3 to success, representing four of ten digits or 40%. Assign 4 through 9 to failure, representing six of ten or 60%. Zero is one of the ten outcomes; accidentally omitting it changes the model. The exact digit labels are arbitrary, but the count assigned to each outcome must match the stated probabilities.

Generate each digit independently with a fair random-digit tool or by drawing from ten equally prepared digit slips, replacing and remixing after every draw. Replacement is essential for the physical-slip method because the probability must remain 40% on every attempt. Drawing slips without replacement would change the later probabilities and simulate a different process.

One trial consists of three digits. Mark the trial as favorable if all three lie from four through nine. A digit from zero through three anywhere in those three positions means the first success occurred before attempt four, so the trial is not favorable. Keep every trial in the record; do not discard an early success because it seems to end the story too quickly.

Illustrative ten-trial record: The triples are 748, 165, 904, 658, 237, 491, 562, 879, 354 and 620. This is a supplied example for practicing classification, not a claim that these digits were generated randomly. Favorable triples are 748, 658 and 879, giving three out of ten or 0.30. Each favorable triple contains only failure digits; every other triple contains at least one success digit.

A theoretical calculation checks the simulation design. Each of the first three attempts fails with probability 0.60 independently, so the event probability is 0.60 × 0.60 × 0.60 = 0.216, or 21.6%. In the digit space there are ten choices at each of three positions, giving 1,000 equally likely triples, and six failure choices at each position give 216 favorable triples. The product and sample-space methods agree.

The illustrative frequency 0.30 differs from 0.216. Ten trials are a small record, so an observed proportion need not equal the model. In an actual simulation, collect many more trials and report both the count and denominator. Increasing trials can help the estimate stabilize, but it does not guarantee exact agreement or a closer result after every additional trial.

The exactly-four event requires a fourth digit from zero through three after three failure digits. Its probability is 0.216 × 0.40 = 0.0864, or 8.64%. Do not use that smaller probability for at least four, because doing so would exclude trials in which success occurs on attempt five or later. Reading the event boundary correctly is as important as generating random numbers.

A simulation can be valid for one model and inappropriate for another. If attempts remove tokens from a small bag without replacement, the success chance changes and independent digits no longer match the process. If success probability is 35%, ten equal digit outcomes are too coarse for an exact assignment; a hundred equally likely two-digit values could assign 35 to success. State what your random device represents and which assumptions it preserves.

Evaluate a simulation through its design before evaluating a surprising result. Check the outcome mapping, independence, trial definition, stopping rule and tally. A perfectly recorded large sample from the wrong model may give a stable but irrelevant answer. A correct model with a small sample may give a rough answer with ordinary variation. These are different reasons for disagreement.

Worked classification: Triple 491 is not favorable for at least four attempts, because digit one signals a success on the third attempt. Triple 748 is favorable because all three are failure digits, so no success has occurred before attempt four. Whether its fourth attempt succeeds is irrelevant to this event.

Worked frequency: Suppose a simulation of 100 trials produces 23 favorable triples. The relative frequency is 23/100 = 0.23. It estimates the model's 0.216 probability and differs by 0.014, or 1.4 percentage points. Do not label 0.23 exact simply because 100 trials were used.

Worked design error: A learner draws three slips from the digit set without replacing them. Once a failure digit is removed, the composition of the remaining slips changes. The later failure chance is no longer six out of ten, so the procedure does not simulate independent attempts with a constant 40% success probability.

Actual simulation: Generate at least 100 three-digit trials using an appropriate independent method. Record the digit source, assignment rule, trial definition, all triples or a reproducible record, favorable count and relative frequency. Compute the running estimate after 20, 50 and 100 trials. Compare with 0.216 and explain variation without claiming that the next outcome must correct a prior imbalance.

Independent performance: Design a second simulation for a related compound question, such as exactly four attempts or at least one success in three. Adjust the event definition and, if needed, trial length. State the model assumptions and check the design with an organized sample space or probability calculation. A partner follows your instructions on several test triples and checks whether every event is classified consistently before you run the full simulation.