AP Statistics – Unit 4 Lesson: Introduction to Probability & Simulation

Topic 4.1 — Probability as Long-Run Relative Frequency

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Key Idea: What Is Probability?

Probability is a number that describes how likely an event is to happen.
But in statistics, we define it more precisely.

Probability = long-run relative frequency.

That means:

• You repeat a random process (like flipping a coin) many, many times.

• You count how often something happens.

• As the number of trials becomes very large, the proportion approaches a stable value.

Example: A Fair Coin

• In one flip, anything can happen: heads or tails.

• In 10 flips, results might be uneven: 7 heads, 3 tails.

• In 100 flips, the proportion of heads will usually be closer to 0.5.

• In 10 000 flips, the proportion will be very close to 0.5.

This shows why probability is not about single outcomes.
It is about patterns over thousands of repetitions.

Unpredictable short-term, predictable long-term

• Short-term outcomes are random.

• Long-term outcomes follow stable patterns.

This idea connects to the Law of Large Numbers:
As the number of trials increases, the relative frequency of an event gets closer to the true probability.

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What Is a Simulation?

A simulation is a way to imitate a real-world random process when actually doing it would be slow, difficult, or impossible.

Simulations use:

• coins

• dice

• spinners

• random number tables

• calculators

• computer programs

• online random generators

Why simulations?

Because they help us estimate probabilities without doing millions of experiments ourselves.

Example: Simulating a Coin Flip

We assume:

• 1 = heads

• 2 = tails

If a calculator generates the random integer 1 or 2, that imitates a coin flip.

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Demonstration: Understanding Randomness

Example: Real coin flips

If someone flips real coins five times, you might see:

• Student A: H T H H T → 3/5 heads = 0.60

• Student B: T T T H H → 2/5 heads = 0.40

• Student C: H H H H H → 5/5 heads = 1.00

Notice how different the results are with only five flips.

What should we expect with more flips?

If we flipped the same coins:

• 50 times → proportions should move closer to 0.5

• 500 times → even closer

• 5000 times → extremely close to 0.5

5. Main Activity: Running a Full Simulation

Run a simulation of 50 coin flips using a random number generator.

Step 1: Set up the simulation

Tell your calculator or Google Sheets to generate random integers 1–100.

Then define:

• Numbers 1–50 = heads

• Numbers 51–100 = tails

This ensures a 50/50 probability, just like a fair coin.

Step 2: Run the simulation

Generate 50 random numbers and count how many are 1–50.

That number = simulated heads.

Example outcome:

• 27 heads out of 50 simulations

• Proportion = 27 ÷ 50 = 0.54

Step 3: Compare class results

Imagine these results:

Student	# Heads	Proportion
A	27	0.54
B	22	0.44
C	25	0.50
D	30	0.60

What patterns do we observe?

• Results vary → short-term randomness

• Most proportions fall near 0.5 → long-run stability

Discuss:

• Why results aren’t identical

• Why results still cluster around 0.5

• How does increasing the number of trials reduce variability?

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6. Application Question

Write your answer clearly.

A company claims a special die is “balanced,” meaning each side (1–6) is equally likely.
Describe a simulation you could run to test this claim. Explain exactly how you would simulate the rolls and how you would interpret the results.

• choose a random number generator

• assign numbers 1–6 to outcomes

• simulate many rolls

• examine proportions

• compare to expected probability = 1/6

-- Put steps 1 - 3 in sentences to answer this question.

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7. Exit Question

Why are simulations useful for estimating probabilities?