Setting Up a Simulation Model

Start With the Chance Process

In What Is a Probability Simulation, a probability simulation was defined as a chance-based process that imitates a real or model situation. In The Law of Large Numbers in Simulations, we saw that the relative frequency of an event tends to settle near its model probability across many repeated trials. Before carrying out those repetitions, we need a clear model: what happens by chance, what outcomes are possible, and what counts as one trial?

Suppose a basketball player makes 70% of free throws over the conditions we want to represent. We will use \(0.70\) as the model probability that the player makes any one shot. A chance process is the chance-based procedure being represented. Here, it is the result of a free-throw attempt: the shot is either made or missed.

This model makes assumptions that should be stated. For now, assume that each shot is independent of the others and has the same probability, \(0.70\), of being made. Those assumptions make the process repeatable in a consistent way. They may be reasonable for a simplified model, but actual shooting probabilities could change with fatigue, pressure, or other conditions.

For one shot, there are two outcomes: a make, which we will write as \(M\), or a miss, written as \(X\). The probability of \(M\) is \(0.70\), and the probability of \(X\) is \(1-0.70=0.30\). In a multi-shot question, the outcome of a trial can be a sequence, such as \(MXX\), rather than just one letter.

Choose a Trial That Matches the Question

A trial is one repetition of the process being simulated. It is not automatically one shot. If the question asks about the result of a single free throw, one trial is one shot. If it asks about a set of three shots, one trial is one three-shot set. The trial definition determines what result gets recorded each time the simulation is repeated.

A useful way to choose the trial is to first state the probability question in ordinary language. Then decide what single, repeatable chunk of the chance process would provide one answer to that question. Finally, define the event that will count as a success within that chunk. In simulation language, “success” means that the event of interest occurs; it does not necessarily mean a basketball shot was made.

A simulation model can be clear even before we decide how to generate its random outcomes. The next tutorial will address assigning random digits to outcomes. Here, the priority is to define what the random digits—or any other chance device—will need to represent.

Sequences Have Probabilities, Not Just Labels

Listing the possible sequences does not mean they are all equally likely. Under the stated model, the probability of an individual sequence depends on how many makes and misses it contains. If a sequence has \(k\) makes in three shots, its probability is \(0.70^k(0.30)^{3-k}\), because the model assumes independent outcomes.

For example, \(MMM\) has three makes, so its probability is \(0.70^3=0.343\). The sequence \(MMX\) has two makes and one miss, so its probability is \(0.70^2(0.30)=0.147\). Under this model, the number of makes determines the probability of an individual sequence; sequences with the same number of makes have the same probability. Order can matter when deciding whether a sequence meets the event rule.

That distinction is useful when the event depends on more than the total number of makes. For “at least two makes,” \(MMX\) and \(MXM\) both qualify. But if the event is “at least two consecutive makes,” \(MMX\) qualifies and \(MXM\) does not. They have the same number of makes and the same individual-sequence probability, but they do not have the same status under that event rule.

Set the Trial Boundary Before Repeating

The right trial depends on the question, not on a universal rule. Imagine instead that the coach asks about the probability of making exactly four shots in a five-shot practice set. One trial should then be one five-shot set, not one shot and not one three-shot set. Each trial has an ordered five-letter sequence, and the event is that exactly four letters are \(M\).

For five shots, there are \(2^5=32\) possible sequences. For instance, \(MMMMX\) and \(XMMMM\) both have exactly four makes and count as successes for this question. There are five successful sequences, one for each possible position of the single miss. The probability of each particular four-make sequence is \(0.70^4(0.30)=0.07203\), under the independence and constant-probability assumptions. The model does not treat the 32 sequences as equally likely.

The definition of one trial also determines the denominator in a later relative frequency. If 100 trials each represent a five-shot set, the simulation has represented 100 sets, or 500 individual shots. For the question about exactly four makes in a set, the relevant relative frequency is the number of successful sets divided by 100—not the number of made shots divided by 500.

Common Mistakes and AP Exam Tips

• Calling one shot a trial when the question concerns a set. If the event is about a three-shot or five-shot set, define one trial as that whole set. Otherwise the recorded result will not directly answer the question.
• Leaving the event vague. “A good set” is not a precise event. Say exactly what counts, such as “at least two makes in three shots” or “at least two consecutive makes in three shots.”
• Recording too little information. If order matters to the event rule, record the sequence, not just the total number of makes. For a consecutive-makes event, \(MMX\) and \(MXM\) cannot be treated as interchangeable.
• Assuming possible sequences are equally likely. Under the 70% make model, \(MMM\) and \(XXX\) do not have the same probability. State the make and miss probabilities and use the independence assumption when considering sequences.
• Confusing a made shot with a successful trial. A simulation success is the event being studied. In a question about exactly four makes in five shots, a successful trial is a whole set with that result, not an individual made shot.
• Forgetting to state assumptions. A full setup says whether shots are assumed independent and whether the make probability stays the same. If those assumptions are not appropriate, the model may need to be revised.

For a clear AP response, connect every part of the model to the question: name the chance process, give its possible outcomes and probabilities, define one trial, and state exactly which trial outcomes count as the event. That setup makes it possible for someone else to repeat the simulation and interpret its relative frequency correctly.