Estimating Probabilities Using Simulation - AP Statistics Study Guide

Quick Summary

This guide will equip you to estimate probabilities of complex random events using simulation. You will learn to design and conduct a simulation by defining components and trials, using tools like random number tables or a calculator, and running the simulation many times. Ultimately, you will be able to use the results of your simulation to draw a conclusion about the approximate probability of an event, understanding that more trials lead to a more accurate estimate.

Key Concepts

When it is difficult or impossible to calculate the theoretical probability of an event, we can use simulation to estimate it. A simulation is a procedure that uses a chance model to imitate a real-world situation.

The Building Blocks of a Simulation

Every simulation is built from a few key parts:

• Component: The most basic event in a simulation whose outcome is determined by a random process. Think of it as one step in the larger process.

  • Example: Flipping a single coin, rolling a single die, or one free-throw attempt by a basketball player.

• Trial: One complete repetition of the simulation, consisting of one or more components. A trial models the full event we are interested in.

  • Example: Flipping a coin 10 times, rolling two dice until you get doubles, or a basketball player attempting 5 free throws.

• Response Variable: The outcome or result that you measure and record at the end of each trial.

  • Example: The number of heads in 10 coin flips, the number of rolls it took to get doubles, or whether the player made at least 4 of their 5 free throws.

The Four-Step Simulation Process

To earn full credit on the AP exam, a simulation must be described using a clear, four-step process.

Step 1: Describe the Model

State the probability model and the tool you will use to imitate one component of the situation.

• Identify the Tool: You can use a random number generator (like on a TI-84 calculator), a table of random digits, a fair coin, or a standard die.

• Assign Outcomes: Clearly assign numbers (or other outcomes) to the events of interest. The assignment must match the known probabilities.

  • Example (75% free-throw shooter): Using two-digit random numbers from 00 to 99:

    • Let 00-74 represent a made free throw.

    • Let 75-99 represent a missed free throw.

• Address Repeats: State whether you will ignore repeated numbers. In situations like drawing cards without replacement, you would ignore repeats. In situations like shooting free throws, repeats are fine because the outcome of one shot doesn't affect the next.

Step 2: Describe a Trial

Explain what constitutes a single trial and what you will record as the response variable.

• Define the Start and End: A trial begins with the first component and ends when the event of interest has occurred or a set number of components have been completed.

  • Example: "A trial will consist of looking at 5 two-digit numbers, representing 5 free-throw attempts."

• Define the Response Variable: State exactly what you will record after each trial.

  • Example: "At the end of each trial, I will record whether the player made 4 or more shots (a 'success') or not (a 'failure')."

Step 3: Conduct Many Trials

Perform a large number of trials and record the results for each. The AP exam will often specify the number of trials to run or provide a random number table to use.

• Show Your Work: Clearly show the random numbers you used and the outcome of each trial. A table is an excellent way to organize this.

• The Law of Large Numbers: This fundamental principle states that as the number of trials increases, the simulated (or empirical) probability will get closer and closer to the true theoretical probability. This is why we must conduct many trials for a reliable estimate. A simulation with only 3 trials is not very trustworthy.

[Image: A line graph showing the cumulative proportion of an event occurring over many trials. The y-axis is "Estimated Probability" and the x-axis is "Number of Trials." The line starts very erratic and gradually flattens out, approaching a horizontal line representing the true probability.]

Step 4: Summarize Results and State a Conclusion

Analyze the results from all your trials and state your conclusion in the context of the problem.

• Calculate the Estimate: The estimated probability is the relative frequency of the "successful" trials.

  • Formula: Estimated Probability = (Number of successful trials) / (Total number of trials)

• State the Conclusion in Context: Always relate your numerical answer back to the original question.

  • Example: "In my simulation of 20 trials, the player made 4 or more shots 6 times. Therefore, the estimated probability of a 75% shooter making at least 4 of 5 free throws is 6/20 = 0.30, or 30%."

Key Vocabulary

• Simulation: A method of modeling random events by using a chance process to generate outcomes that are consistent with a real-world situation, allowing for the estimation of probabilities.

• Component: The most basic, single random event that makes up a trial. For example, one roll of a die.

• Trial: One full repetition of a simulation, which consists of one or more components, that models the event of interest.

• Response Variable: The outcome of a trial that is recorded and analyzed.

• Law of Large Numbers: The statistical principle stating that as the number of trials in a simulation increases, the proportion of times a specific outcome occurs will approach its true theoretical probability.

• Random Digits Table: A table of digits (0-9) where each digit is equally likely to appear in any position, used as a tool for conducting simulations.

Calculator Tech (TI-84)

The primary function for simulation on the TI-84 is the random integer generator.

Function:$randInt()$

This function generates random integers within a specified range.

Path:MATH -> PRB -> 5:randInt()

Usage:

1. $randInt(lower,upper)$: Generates a single random integer between $lower$ and $upper$, inclusive.

   • Example: To simulate rolling one standard six-sided die, you would use $randInt(1,6)$. Each time you press ENTER, it will generate a new random integer from 1 to 6.

2. $randInt(lower,upper,n)$: Generates $n$ random integers between $lower$ and $upper$, inclusive. This is useful for running multiple components at once.

   • Example: To simulate five rolls of a die, you would use $randInt(1,6,5)$. The calculator will return a list of five numbers, like $4,1,1,6,3$.

Seeding the Calculator (Optional but good practice):

To ensure that everyone in a class gets the same sequence of "random" numbers for a given simulation, you can "seed" the random number generator.

1. Type a unique number (like your student ID or a number provided by your teacher).

2. Press STO->.

3. Press MATH -> PRB -> 1:rand.

4. Press ENTER. The calculator is now seeded.

How to Show Work on the FRQ

Simulation questions on the AP exam are graded based on the clarity and completeness of your four-step process. Use this template to ensure you earn full credit.

Question Template: A company is giving away one of five different toys in its cereal boxes. Assuming the toys are distributed equally, design and conduct a simulation to estimate the probability that you will have to buy 10 or more boxes to collect all five toys.

FRQ Response Template:

Step 1: State the Model

I will use a random number generator to simulate buying cereal boxes. The five toys are equally likely, so each has a 1/5 = 20% chance of being in any given box.

• Tool: I will use $randInt(1,5)$ on a TI-84 calculator.

• Assignment:

  • 1 = Toy 1

  • 2 = Toy 2

  • 3 = Toy 3

  • 4 = Toy 4

  • 5 = Toy 5

• Repeats: I will allow repeated numbers, as you can get the same toy multiple times.

Step 2: Describe One Trial

A trial consists of generating random integers from 1 to 5 until all five unique integers (1, 2, 3, 4, and 5) have been generated. The response variable for each trial is the number of integers (boxes) it took to get all five. I will record whether this number is 10 or more (a "success").

Step 3: Conduct the Trials

I will conduct 10 trials.

(Here, you would show your work clearly. A table is best.)

(Fill in all 10 trials based on your calculator's output.)

Step 4: State the Conclusion

In my simulation of 10 trials, I found that 3 of the trials required 10 or more boxes to collect all five toys.

Based on this simulation, the estimated probability of needing to buy 10 or more boxes is 3/10 = 0.30.

Practice Problems

Problem 1:

A basketball player makes 80% of her free throws. Assume each shot is independent. She is in a situation where she will shoot 3 free throws. Use the provided line of random digits to estimate the probability that she makes at least 2 of the 3 shots. Conduct 10 trials.

Random Digits: 81486 69487 60513 09297 00412 71238 27649 39950

Solution:

Step 1: State the Model

I will use the provided line of random digits to simulate the 3 free-throw attempts. The player has an 80% chance of making a shot.

• Tool: I will use two-digit numbers from the random digits table.

• Assignment:

  • Let 00-79 represent a made shot.

  • Let 80-99 represent a missed shot.

• Repeats: I will allow repeated numbers because the shots are independent.

Step 2: Describe One Trial

A trial will consist of reading three consecutive two-digit numbers from the table. For each trial, I will count the number of made shots. The response variable is whether the number of made shots is 2 or more (a "success").

Step 3: Conduct the Trials

I will conduct 10 trials, starting from the beginning of the provided line of random digits.

Correction for a realistic FRQ: An FRQ would provide enough digits. Let's assume the line continues. Let's re-group the digits to be clearer: $\mathrm{81486694876051309297...}$

Let's re-run the trials properly:

After 10 simulated trials: I found 8 successful trials.

Step 4: State the Conclusion

In my simulation of 10 trials, there were 8 trials where the player made at least 2 shots. Based on this simulation, the estimated probability of this player making at least 2 of 3 free throws is 8/10 = 0.80.

Problem 2:

A student is taking a 5-question multiple-choice quiz. Each question has four options (A, B, C, D), only one of which is correct. The student has not studied and decides to guess randomly on every question. Design and conduct a simulation using your calculator to estimate the probability that the student passes the quiz by getting at least 3 questions correct. Run 20 trials.

Solution:

Step 1: State the Model

I will use my calculator's random number generator to simulate guessing on the 5-question quiz. For each question, the probability of guessing correctly is 1/4 or 25%.

• Tool: I will use $randInt(1,4)$ on my calculator.

• Assignment:

  • Let 1 represent a correct answer.

  • Let 2, 3, 4 represent an incorrect answer.

• Repeats: I will allow repeated numbers, as guessing correctly on one question does not affect the others.

Step 2: Describe One Trial

A trial will consist of generating 5 random integers from 1 to 4, representing the 5 questions on the quiz. The response variable will be the number of 1s (correct answers) in the set of 5 integers. I will record whether this number is 3 or more (a "pass").

Step 3: Conduct the Trials

I will conduct 20 trials. For each trial, I will use $randInt(1,4,5)$.

(After completing all 20 trials, let's assume we found 2 trials resulted in a "Yes".)

Step 4: State the Conclusion

In my simulation of 20 trials, the student passed the quiz (got 3 or more correct) in 2 of the trials. Based on this simulation, the estimated probability of passing the quiz by guessing is 2/20 = 0.10, or 10%.

Common Mistakes to Avoid

• Incomplete Model Description: Forgetting to explicitly link the random numbers to the outcomes. It's not enough to say "I'll use numbers 1-4." You must state what each number represents (e.g., "1 = correct, 2-4 = incorrect"). Also, failing to state how you will handle repeats (even if it's just to say they are allowed) can lose points.

• Using the Wrong Tool: If an FRQ provides a random number table, you must use it. Do not use your calculator's $randInt()$ function instead. Your work must be reproducible from the provided materials.

• Confusing Component with Trial: A trial is the full process you are interested in (e.g., shooting 3 free throws). A component is a single part of that (e.g., shooting 1 free throw). Make sure your description of a trial is not just a description of a component.

• Making a "Naked" Conclusion: Do not just state a final probability. Your conclusion must be in the context of the problem. Instead of "The probability is 0.10," write "Based on my simulation, the estimated probability of passing the quiz by guessing is 0.10."

• Not Using "Enough" Trials: A simulation with very few trials (e.g., 2 or 3) is not reliable. While an FRQ will specify the number of trials, remember the Law of Large Numbers implies that more trials lead to a better estimate. Do not claim your estimate is the "true" probability; always refer to it as the "estimated probability based on the simulation."