AP Statistics · Topic 2.3 · Unit 2
AP Stats 2.3: Estimating Probability by Simulation
By Jude Wallis · Published
Topic 2.3 in the Fall 2026 AP Statistics course estimates probabilities with simulation. You model a random process, run many trials, and use the long-run relative frequency of an event as its estimated probability. The law of large numbers says that estimate settles toward one value as trials grow.
AP Statistics: Unit 2 (topics 2.3). In the Fall 2026 AP Statistics course, estimating probabilities using simulation is Unit 2 Topic 2.3, aligned to skill 3.C.
What topic 2.3 covers
Topic 2.3 introduces probability through simulation before any formulas arrive. The objective is to estimate probabilities using simulations, which builds the intuition the rest of Unit 2 formalizes.
Three words anchor the topic. A random process generates results determined by chance, an outcome is the result of one trial of that process, and an event is a collection of outcomes you care about.
How a simulation estimates a probability
A simulation models a random event so that the simulated outcomes closely match real-world outcomes. You associate every possible outcome with a value to be determined by chance, run many trials, and record the counts of the simulated outcomes along with the total number of trials.
The probability of an outcome or event is its long-run relative frequency, meaning its relative frequency over a large number of trials. So the relative frequency you observe in the simulation, the count of successes divided by the number of trials, estimates the actual or true probability. Random digits, coins, dice, or a calculator can all supply the chance values.
The law of large numbers
The estimate improves as you run more trials. The law of large numbers states that for independent trials, as the number of trials increases, the long-run relative frequency of an outcome or event gets closer and closer to a single value.
That single value is the true probability. A run of ten trials can land well off the mark, while thousands of trials pin the estimate down, which is why more simulated trials give a better estimate. This is a statement about long-run stability, not about the shape of a distribution, so it is distinct from the central limit theorem even though both describe what happens with large samples. You can watch the effect in the sampling distribution simulator.
Designing a good simulation
A trustworthy simulation rests on three design choices you state before running it. First, decide how a chance device maps to outcomes, such as letting digits 0 through 6 stand for a make and 7 through 9 for a miss when a shooter succeeds 70% of the time.
Second, define exactly what one trial is, including how many chance values it uses and what counts as a success. Third, run many trials and record both the number of successes and the total number of trials. Writing these rules down first is what makes the resulting estimate defensible, and it matches the exam expectation to show the structure of a probability calculation rather than only its answer.
Estimating a free-throw streak with random digits
A player makes 70% of free throws. Estimate the probability she makes at least 4 of her next 5 attempts by simulation. Let digits 0 through 6 stand for a make (7 of 10) and digits 7 through 9 for a miss, reading five digits per trial.
Trial 1 uses 4 8 1 9 3, giving makes at 4, 1, 3, so 3 makes: not a success.
Ten trials of five digits produce these make-counts: 3, 4, 4, 3, 5, 2, 5, 3, 5, 2.
A trial succeeds when it has at least 4 makes; the successes are trials 2, 3, 5, 7, and 9.
Count successes over trials: .
The simulation estimates the probability at about . By the law of large numbers, running thousands of trials would tighten this estimate toward the true value.
Frequently asked questions
How does a simulation estimate a probability?
You model the random process, assign every outcome a value set by chance, and run many trials while recording successes and the total number of trials. The relative frequency of the event, successes divided by trials, estimates the true probability because probability is defined as long-run relative frequency.
What is the law of large numbers?
It states that for independent trials, as the number of trials increases, the long-run relative frequency of an outcome gets closer and closer to a single fixed value, the true probability. It explains why more simulated trials give a more reliable probability estimate.