Simulation is a powerful method for estimating probabilities when an exact calculation is difficult or time consuming. Instead of listing every possible outcome, a computer or calculator generates many random trials that imitate the real situation. The fraction of trials that produce the event of interest becomes an estimate of its probability.
This matters because simulation is used in science, engineering, finance, games, and everyday decision making.
Key Facts
- Estimated probability = number of successful trials / total number of trials
- If X is the number of successes in n trials, then p_hat = X / n
- For a fair random digit generator, each digit 0 through 9 has probability 1/10
- The law of large numbers says p_hat tends to get closer to the true probability p as n increases
- For repeated independent trials, standard error is approximately sqrt(p_hat(1 - p_hat) / n)
- A 95% rough margin of error for a simulated proportion is about 2sqrt(p_hat(1 - p_hat) / n)
Vocabulary
- Simulation
- A simulation is a model that uses repeated trials to imitate a random process and study its outcomes.
- Random number generator
- A random number generator is a tool that produces numbers designed to behave unpredictably according to a specified rule.
- Trial
- A trial is one complete run of a simulated random process.
- Estimated probability
- An estimated probability is the proportion of simulated trials in which the event of interest occurs.
- Convergence
- Convergence is the tendency of simulation results to approach the true value as the number of trials increases.
Common Mistakes to Avoid
- Using too few trials, which makes the estimate highly variable and easy to misinterpret. Increase the number of trials to reduce random fluctuation.
- Assigning random numbers unevenly, which changes the probability model being simulated. Make sure each outcome gets a number range proportional to its true probability.
- Treating the simulated estimate as exact, which ignores sampling variability. Report it as an estimate and consider a margin of error.
- Forgetting to make trials independent, which can bias the results if one trial affects the next. Reset the conditions after each trial unless dependence is part of the model.
Practice Questions
- 1 A simulation of 500 spins of a fair spinner gives 128 lands on blue. Estimate the probability of landing on blue.
- 2 A random number method uses digits 0 through 9. Digits 0, 1, and 2 represent a win, while digits 3 through 9 represent a loss. In 800 trials, how many wins would you expect?
- 3 Two students simulate the same probability. Student A uses 50 trials and gets 0.62, while Student B uses 5000 trials and gets 0.54. Explain which estimate is usually more reliable and why.