Describing a Simulation in Words

Turn a Basketball Question into a Simulation Description

In Setting Up a Simulation Model, you learned to specify a chance process, its possible outcomes, the event of interest, and what counts as one trial. This tutorial focuses on putting those choices into a complete description in words. The setting is a basketball player who makes a free throw with probability 0.70 on each attempt. We want to estimate the probability that the player makes at least three consecutive free throws in a sequence of eight attempts.

The challenge is not just to say “simulate free throws.” A clear description must explain how random outcomes stand for makes and misses, what one complete trial contains, how many times to repeat that trial, and exactly what to record. If any of those pieces is missing, another student may not be able to carry out the same simulation or decide consistently whether a trial counts as a success.

Four Parts of a Complete Description

For this problem, a trial is not a single shot. One trial must represent the full situation being studied: eight free throws by the player. The event is that at least three of those eight results are makes in a row. A trial succeeds if that event happens anywhere in the eight shots, whether or not there is another streak as well.

For a 70% make probability, random digits 0 through 9 provide ten equally likely labels. Assign seven digits to a make and the remaining three to a miss. For instance, digits 0 through 6 can represent makes, while 7 through 9 represent misses. Each digit is equally likely, so this assignment gives a 7-out-of-10, or 70%, chance of a make.

Use a new random digit for each of the eight shots, then use eight more for the next trial. On a calculator, a group of eight digits can be generated with a command such as \(\operatorname{randInt}(0,9,8)\). Each generated digit represents one shot, and each group of eight represents one trial. This models the stated chance for every shot; the model also treats shot outcomes as independent. As discussed in Setting Up a Simulation Model, the simulation needs to preserve the chance process described in the question.

Decide What Counts as One Successful Trial

A recording rule should be precise enough that two people examining the same sequence would reach the same decision. One convenient way to check a sequence is to translate each digit into M for a make or X for a miss, then scan the eight results in order. A success requires at least one block of three consecutive M’s. A miss breaks a streak, but the count of consecutive makes can begin again after that miss.

The word “at least” matters. A sequence with four consecutive makes contains a run of three consecutive makes, so it qualifies. The event does not require exactly three makes in a row, and the player does not have to make the first three shots. The streak can begin at any shot position as long as three consecutive shots are made.

Repeat the Same Trial and Record Consistently

After classifying a trial, start the next trial with a fresh group of eight random digits. Do not carry a streak from the end of one trial into the beginning of the next. For example, if shot 8 is a make in one trial and shot 1 is a make in the next, those shots are not consecutive attempts within the same eight-shot sequence.

The number of repetitions is the number of complete eight-shot sequences simulated, not the total number of individual shots. If the plan calls for 200 repetitions, it generates \(200 \times 8=1{,}600\) digits in all, grouped into 200 trials. The event is evaluated separately within each group of eight.

What a Clear AP-Style Description Sounds Like

A strong response reads like a set of instructions another person could follow without needing to guess. Include the chance assignment in enough detail to verify the probabilities. Define the unit that makes up a trial, and identify what counts as a success within that trial. Then state the number of repetitions and the value to record for every repetition.

For this basketball problem, a concise but complete description could be: “Use random digits 0–9, with 0–6 representing a made free throw and 7–9 representing a missed free throw. Generate eight digits for one trial, representing eight shots. A trial is a success if any three or more consecutive digits represent makes. Repeat with a fresh group of eight digits for each of 1,000 trials, and record yes or no for whether the streak occurs in each trial. Estimate the probability by dividing the number of yes results by 1,000.”

Notice how the description connects the model to the event. It does not merely say “repeat the simulation 1,000 times”; it says what is repeated. It also does not leave the reader to decide whether a streak must start at the beginning or whether a longer run counts.

Common Mistakes and AP Exam Tips

• Calling one shot a trial. The question concerns a streak in eight attempts, so one trial must include all eight shots. A single shot cannot show whether the eight-shot streak event occurred.
• Failing to match the assignment to the probability. Giving makes only five of ten digits would model a 50% chance, not 70%. State exactly which digits mean make and which mean miss, and check their proportions.
• Not saying where the streak can occur. The three makes can occur anywhere in the eight shots. A full-credit rule says “any three or more consecutive makes,” rather than just “make three free throws.”
• Counting shots instead of successful trials. The estimate’s numerator is the number of eight-shot trials that meet the event, not the total number of made shots. Each trial receives one yes-or-no record.
• Joining separate trials together. A run cannot begin in one trial and finish in the next. Each new group of eight is a separate sequence, evaluated independently.
• Leaving the repetition count vague. Say a specific number, such as 1,000 trials, and explain that each repetition uses a new group of eight random outcomes.
• Recording an unclear result. “Track the streak” could mean several things. State whether to record yes/no, and define the yes rule so another person can apply it consistently.

For full credit, use details that preserve the situation being modeled: the 70% chance for each shot, eight shots in each trial, a streak of at least three consecutive makes, fresh random outcomes for each repetition, and a clear record of whether the streak occurred. The goal is a repeatable description, not just a general statement that randomness is involved.