Introduction to Probability and Chance

Using Simulation to Model Reality

Because real-world scenarios can be complex, mathematicians use simulation to model random behavior. A simulation is a method of imitating a real-world process using a mathematical model so that the simulated outcomes are consistent with real-world outcomes.

To conduct a successful simulation, one must follow a structured process:

1. Describe the trial: Define how to use a random process to perform one single repetition (a trial) and determine what will be recorded.
2. Perform many trials: Repeat the process numerous times to gather data.
3. Analyze the results: Use the collected data to answer the original question of interest.

Example: The "1 in 6 Wins" Claim

Imagine a soda company claims that "1 in 6" bottles contains a winning cap. To test this, a class of 30 students buys one bottle each, and only 2 students win. Is the company lying?

To find out, we can use a six-sided die as a model: rolling a 6 represents a "winner," while rolling 1 through 5 represents "try again". By rolling the die 30 times (one trial) and repeating this many times, we can create a dotplot to see how often "2 or fewer winners" occurs by pure chance.

Determining Plausibility and Convincing Evidence

One of the most important uses of probability is determining plausibility—whether an observed result is a likely outcome of random chance or if it provides convincing evidence that a claim is inaccurate.

Consider a NASCAR promotion where a fan must collect 5 different driver cards. The company claims all 5 cards are equally likely. If the fan takes 23 boxes to get the full set, we can run a simulation to see how often that happens.

If the simulation shows that the probability of needing 23 or more boxes is nearly $0%$, then the fan's experience provides convincing evidence that the cards are not equally likely. However, if the probability is high (e.g., $9%$), the result is considered plausible, meaning the outcome could have happened by chance alone, and we do not have enough evidence to call the company a liar.

AP® Exam Tip

When describing a simulation using random digits, always provide a clear enough description so that someone else could replicate your exact results. Ensure every label is the same length. For example, if labeling students 01 to 95, use two-digit pairs. If sampling without replacement, explicitly state that repeated numbers should be ignored.