Difficulty: Beginner | Prerequisites: Basic understanding of relative frequency and data collection from Units 1 – 4.
This is the start of the probability unit in AP Statistics, and it underpins everything that comes later: inference, confidence intervals, significance tests. You need to understand what probability means (long-run relative frequency), why individual trials are unpredictable, and how simulations let you estimate probabilities when formulas are impractical. If you skipped the exploratory data analysis units, go back and review relative frequency first.
Probability describes what happens in the long run, not on any single trial. The Law of Large Numbers says observed proportions settle towards the true probability as trials increase. Simulation is a hands-on method for estimating probabilities by using random devices (dice, random number generators) to model a real situation many times.
Probability
The long-run relative frequency of an outcome. If you repeat a random process over and over, the proportion of times the outcome occurs approaches its probability.
In simple terms, probability tells you "how often would this happen if I could repeat the process thousands of times."
Law of Large Numbers
As the number of trials of a random process increases, the observed proportion of a particular outcome gets closer and closer to the true probability of that outcome.
Think of it as: short runs are noisy, long runs are stable. Flip a coin 10 times and you might get 70% heads. Flip it 10,000 times and you will be very close to 50%.
Simulation
The imitation of a random process using a device (die, coin, random number table, random number generator) to estimate a probability.
In simple terms, when you cannot calculate a probability with a formula, you act it out many times and see how often the outcome occurs.
Random number table
A pre-printed list of digits (0 – 9) in which each digit is equally likely and each position is independent. Used to carry out simulations by assigning digit ranges to outcomes.
Think of it as a low-tech random number generator you can use on paper during an exam.
Probability is a long-run concept. Saying "the probability of heads is 0.50" means that across a very large number of flips, about half will be heads.
On any single flip, or even a short run of flips, the observed proportion can be far from 0.50. That is normal, not evidence that the coin is unfair.
Example: if the probability that a randomly selected U.S. adult drinks coffee on a given day is 0.56, that does not mean exactly 56 out of 100 randomly selected adults will drink coffee. It means, across many samples of 100, the average proportion will be close to 0.56.
A probability of 0 means the outcome is impossible.
A probability of 1 means the outcome is certain.
A probability near 0 (e.g. 0.001) means the outcome is very unlikely but will happen occasionally in a long sequence.
A probability above 0.5 (e.g. 0.6) means the outcome occurs more often than not.
A probability near 1 (e.g. 0.99) means the outcome is almost certain but not guaranteed.
Past outcomes of independent events do not affect future outcomes.
Example: a couple has had seven girls in a row. Their doctor says a boy is "due." The doctor is wrong. Each birth is still approximately 50/50 for boy or girl, regardless of what happened before.
The Law of Large Numbers works by dilution (future results swamp past ones over time), not by compensation (the process does not "remember" it owes you a different result).
A well-designed simulation will:
Accurately model the real-world probabilities (e.g. use a six-sided die if the probability is 1/6).
Clearly define what counts as a "success" and a "failure."
Run enough trials that the estimated probability is stable.
Pepsi claims "1 in 6 wins a prize." A class of 30 students buys one bottle each and only 2 win. Is that suspicious?
To investigate: roll a six-sided die for each of the 30 students. Let one face (say, 1) represent "winner" and the other five represent "loser."
Roll the die 30 times (one trial). Count how many winners. Repeat many trials.
After many trials, check how often you got 2 or fewer winners. If it happens fairly often (say 10 – 15% of the time), the result is plausible under the claimed probability. If it almost never happens, you have evidence the claim may be wrong.
Assign numbers to outcomes. For 95 students numbered 01 – 95, read two-digit pairs from the table. Skip 96 – 99 and 00 (or assign them as needed).
For each trial, read enough pairs to select the required number of individuals. Skip repeats if sampling without replacement.
Record the result for each trial, then repeat.
If you want to know whether an observed outcome is suspicious, check what proportion of your simulated trials produced a result as extreme as (or more extreme than) the one observed.
A large proportion (e.g. 9 out of 100 trials, or 9%) suggests the outcome could plausibly happen by chance, so there is not strong evidence of foul play.
A very small proportion (e.g. 1 out of 100) would suggest the outcome is unlikely under the assumed model, and something else may be going on.
Probability as relative frequency:
P(outcome) = (number of times outcome occurs) / (total number of trials), as the number of trials grows large.
No new algebraic formulas in these lessons. The emphasis is conceptual and simulation-based.
Insurance companies set premiums using probability: they cannot predict which individual policyholder will file a claim, but the Law of Large Numbers lets them estimate the proportion of claims across thousands of customers with high accuracy. Simulations are used in engineering, finance, and sports analytics to estimate probabilities that are too complex for closed-form solutions.
Students often think probability predicts what will happen in a small sample. It does not. Probability is a long-run property, and short runs are inherently variable.
Students confuse the Law of Large Numbers with the gambler's fallacy. The law says proportions stabilise over many trials; it does not say the process "corrects" itself after a streak.
Students sometimes think a simulation with only a few trials is sufficient. You need many trials (typically at least 50 – 100) for the estimated probability to be useful.
Students forget to explain what their simulation components represent. On the AP exam, you must state explicitly what each random device outcome maps to (e.g. "1 on the die = winner, 2 – 6 = loser").
⚠️ The AP exam will ask you to "interpret a probability as a long-run relative frequency." Use this phrasing: "If we select many, many [subjects] at random, about [X%] of them will [outcome]."
⚠️ Free-response simulation questions require you to describe the model, define what constitutes a trial, state what you are recording, and explain how you use the results. Missing any of these steps loses marks.
⚠️ When asked "does the result provide convincing evidence," you must reference the simulation proportion. Do not just say "yes" or "no" without stating the estimated probability.
⚠️ The gambler's fallacy appears regularly. If a question describes a streak and asks if the next outcome is affected, the answer is almost always no (assuming independence).
True or false: If the probability of rain on any given day is 0.30, then in a 10-day stretch exactly 3 days will have rain.
Fill in the blank: The __________ says that as the number of trials increases, the observed proportion approaches the true probability.
True or false: After flipping 8 heads in a row with a fair coin, the probability of heads on the 9th flip is less than 0.50.
Fill in the blank: A simulation should use a random device that accurately models the __________ of the real situation.
True or false: Getting 2 winners out of 30 when the probability is 1/6 is impossible if the company is telling the truth.
Answers: 1. False (probability does not guarantee exact counts in small samples). 2. Law of Large Numbers. 3. False (each flip is independent; still 0.50). 4. Probabilities. 5. False (unlikely outcomes can still happen by chance).
Q: The probability that a randomly selected voter in a certain town supports a bond measure is 0.62. Interpret this probability as a long-run relative frequency.
A: If we randomly select voters from this town over and over again, in the long run about 62% of them will support the bond measure.
Q: A class conducts a simulation to test whether a coin is fair. In 200 simulated sets of 50 flips each, only 3 sets had 35 or more heads. The class flipped an actual coin 50 times and got 36 heads. What should they conclude?
A: In only 3 out of 200 simulated trials (1.5%) did a fair coin produce 35 or more heads. Since 36 heads is in that rare region, this provides convincing evidence that the coin may not be fair.
Q: Describe how you would use a random number generator to simulate the cereal-box card-collecting problem where each of 5 cards is equally likely.
A: Generate random integers from 1 to 5, where each number represents one of the 5 cards. Keep generating numbers until all 5 distinct values have appeared. Record how many numbers you generated. Repeat for many trials, then use the results to estimate probabilities about how many boxes are needed.
Q: Explain why the gambler's fallacy is a fallacy, using the Law of Large Numbers.
A: The Law of Large Numbers says the overall proportion of outcomes converges to the true probability over many trials, but this happens because future trials dilute past results, not because the process compensates for a streak. Each independent trial has the same probability regardless of what happened before.
This material connects directly to sampling distributions (Unit 7), where you will see the Law of Large Numbers at work: as sample size increases, the sample proportion gets closer to the population proportion. Simulation also reappears in hypothesis testing (Unit 9), where you simulate data under a null hypothesis to see whether an observed result is unusual.
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