AP Statistics Unit 4 Probability & Random Variables

Topics 4.1–4.2: Random Patterns & Estimating Probabilities Using Simulation

Introducing Randomness and Simulation
Video Follow-Along Worksheet
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Learning Objectives:

Key Vocabulary

Random Process A situation where all possible outcomes are known, but individual outcomes are unpredictable
Outcome The result of a single trial of a random process
Event A collection of outcomes from a random process
Simulation A way to model random events such that simulated outcomes closely match real-world outcomes
Empirical Probability Probability estimated by performing many trials and calculating relative frequency
Law of Large Numbers As the number of trials increases, simulated probabilities get closer to the true probability

Topic 4.1: Random and Non-Random Patterns

[Video: ~5 min]
1. A is a situation where we know all the possible outcomes, but we do not know what the will be.
2. Although individual outcomes are uncertain, there is a sense of that occurs in the .

Give It a Chance! Activity

Without flipping a coin, write down a sequence of 100 "random" coin flips (H or T) that you think could come from a fair coin:

3. In the video's actual 100 coin flips, there were heads and tails.
4. Even if you see 5 tails in a row, this does not mean the next flip is "due" to be heads. Each flip is .
5. The longest streak in the video was tails in a row. This may seem non-random, but such streaks are actually in random data.
Key Insight: Patterns in random data (like streaks) may appear non-random, but they are a natural part of random processes. How likely is a streak of 8+? Stay tuned for Topic 4.2!

Video 1: Estimating Probabilities Using Simulation (Topic 4.2)

[Video: ~6 min]
6. An is the result of a single trial of a random process.
7. An is a of outcomes. (Example: rolling a prime number = {2, 3, 5})
8. To find the probability of getting a streak of 8+ heads or tails in 100 flips from just one trial gives us %. This is problematic because we only performed trial.
9. Due to in random processes, we need to perform trials to get a reliable estimate.
10. is a way to model random events such that simulated outcomes closely match outcomes.
11. In a simulation of 50 sets of 100 coin flips, a streak of 8+ occurred in out of 50 trials, giving an estimated probability of about %.

Video 2: Estimating Probabilities Using Simulation (Topic 4.2)

[Video: ~8 min]
12. The states that simulated probabilities tend to get to the true probability as the number of trials .
13. In the video's "fair" coin simulation, after 1,000 flips the proportion of heads was around %. This suggests the coin is .
Steps to Conduct a Simulation:
  1. Describe how random digits will imitate the trial (assign digits to outcomes)
  2. Determine what will be recorded from each trial
  3. Perform many trials of the simulation
  4. Calculate the of successful trials
Sharpshooter Example: A basketball player makes 82% of her free throws. To break the team record, she must make 16 consecutive shots. Design a simulation to estimate the probability of breaking the record.
14. Step 1 - Assign digits:
Using a random number generator (1-100):
• Numbers to represent a
• Numbers to represent a
15. Step 2 - Stopping rule:
Continue selecting random numbers until a occurs. Count the number of .
16. Results: Out of 200 simulated trials, resulted in a streak of 16+ made shots.
Estimated probability: / = %
17. If using a table of random digits instead, you must use -digit numbers (01-82 = make, 83-99 and 00 = miss).

Reflection Questions

[AI-Graded]
R1. Explain why your "fake" coin flip sequence (from the activity) might look different from truly random data. What patterns do humans tend to avoid that actually appear in random sequences?
R2. A friend says: "I've flipped tails 5 times in a row, so the next flip is almost certain to be heads." Explain why this reasoning is flawed, using the concept of independent events and the Law of Large Numbers.

Exit Ticket

Design a simulation: A multiple choice test has 5 questions, each with 4 options (A, B, C, D). A student guesses randomly on every question. Design a simulation to estimate the probability of getting at least 3 correct by guessing.

Class Responses (Esc to close)

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