Key takeaway: A simulation is a model that uses a random device, such as a die or random numbers, to imitate a real chance situation and estimate its probability from many repeated trials.
Grade 77.SP.C.8cProbability and Counting
Definition of Simulation
A simulation stands in for something that is hard, slow or impossible to try for real. Suppose a cereal company hides one of 4 toys in each box, all equally likely. Buying hundreds of boxes to see how many you need is silly, so you roll a 4-sided die instead. Each roll plays the part of opening one box.
To build one, match each outcome of the real event to an outcome of the device, keep the same chances, and decide what counts as one trial. Then repeat the trial many times and record the results. The estimate improves as the number of trials grows, though it will never be perfectly exact.
Formula and notation
A simulation reports an experimental estimate from its trials:
\[P(\text{event}) \approx \frac{\text{trials in which the event occurred}}{\text{total trials}}\]
When the answer is a count, such as boxes needed, the average of the trials is used.
Key facts
- Choose a device that matches the real chances, such as a 4-sided die for 4 equal outcomes.
- One trial must reproduce a whole run of the real situation.
- Dozens or hundreds of trials give a steadier estimate than just a few.
- Random number generators let a computer do thousands of trials quickly.
Simulation example
Each cereal box holds one of 4 toys, all equally likely. Aaliyah rolls a 4-sided die until all four numbers have appeared, counting each roll as one box. Her five trials took 7, 9, 6, 12 and 11 rolls. Estimate how many boxes a collector needs.
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Add the five trial results: \(7 + 9 + 6 + 12 + 11 = 45\).
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Divide by the number of trials: \(45 : 5 = 9\).
The simulation estimates about 9 boxes are needed to collect all 4 toys.
Common mistake
Watch out
Aaliyah ran just one trial, got 7 rolls, and announced that every collector needs exactly 7 boxes.
Why it is wrong
One trial is only one random outcome, and her own five trials ranged from 6 to 12. A simulation needs many repetitions so that the extremes average out. With more trials the estimate would settle near the true long-run average, which is about 8.3 boxes.
Related terms and lessons
- Grade 7 · Probability of Compound EventsLessonWorksheetQuiz
- Algebra 2 · Statistical Experiments, Simulation and DecisionsLessonWorksheetQuiz
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