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Design Simulations for
Compound Events

Learn how to model real-world probability by designing simulations that generate frequencies for compound events.

7th Grade Math

What Is a Simulation?

A simulation is an experiment that models a real-world situation using simple tools: coins, dice, spinners, random number generators, or cards. Instead of observing the actual event (which may be costly or slow), you imitate it and record outcomes.

What Are Compound Events?

Simple Event
A single outcome: rolling a \(4\) on one die, or flipping heads once.
Compound Event
Two or more simple events combined: rolling a \(4\) and flipping heads, or drawing two red cards in a row.

Why Simulations Matter

Sometimes you can calculate a theoretical probability. But when the situation is complex, messy, or hard to model mathematically, a simulation lets you estimate probabilities by running many trials and counting results.

Estimate
Generate experimental data when theory is hard.
Verify
Check if your calculated probability matches real-world data.
Predict
Forecast how often a compound event occurs over many trials.

The Core Idea

Real-World
Compound Event
"Find 2 type-A donors"
Design a
Simulation
Assign numbers,
run trials
Analyze
Frequencies
Estimate
probability

Choosing a Simulation Tool

Match the Tool Coin Flip 2 equally likely outcomes (50/50) Number Cube 6 equally likely outcomes Random Digits 10 outcomes (0-9), flexible Spinner Custom unequal probabilities Rule: The tool's outcomes must match the real event's probability structure.

Theoretical vs. Experimental

Theoretical Experimental (Simulation)
Based on Mathematical reasoning Running actual trials
Formula \(\dfrac{\text{favorable}}{\text{possible}}\) \(\dfrac{\text{successes}}{\text{trials}}\)
Accuracy Exact (if model is correct) Approximate (improves with more trials)
Best for Simple, well-defined events Complex or messy compound events

How to Design a Simulation

1
Identify the compound event. What real-world outcome are you modeling?
2
Choose a simulation tool. Match the tool's outcomes to the event's probability (coins for 50/50, dice for \(\frac{1}{6}\), digits 0-9 for tenths, etc.).
3
Define one trial. Describe exactly what actions make up a single trial and what counts as "success."
4
Run many trials. Repeat (at least 20-50 times) and record every result.
5
Analyze the data. Calculate experimental probability: \(\frac{\text{successes}}{\text{total trials}}\).

Example 1: Blood Donors

Problem: About \(40\%\) of people have type A blood. You need 2 type-A donors. How many people must you screen to find 2?

1
Tool: Random digits 0-9. Let digits 0, 1, 2, 3 = type A (that's \(4\) out of \(10 = 40\%\)). Digits 4-9 = not type A.
2
One trial: Generate random digits until you get two digits in {0,1,2,3}. Count how many digits that took.
3
Repeat for 30+ trials and find the average number of people screened.

Example 1: Sample Trials

Trial 1 7 2 5 8 1
→ 5 people screened
 Trial 2 3 0
→ 2 people screened
 Trial 3 6 9 4 1 8 0
→ 6 people screened
Average after 3 trials: \( \frac{5 + 2 + 6}{3} \approx 4.3 \) people

Example 2: Cereal Prizes

Problem: A cereal box contains \(1\) of \(5\) different prizes (equally likely). How many boxes must you buy to collect all 5 prizes?

1
Tool: A number cube (die with faces 1-5, ignore 6). Or use digits 1-5 from a random number generator.
2
One trial: Keep rolling (or generating) until all five numbers {1, 2, 3, 4, 5} appear. Count the total rolls.
3
Repeat 30+ times. Average the totals to estimate the expected number of boxes.

Example 3: Free Throws

Problem: A basketball player makes \(70\%\) of free throws. What is the probability she makes both shots in a 2-shot foul?

Simulation Design
Digits 0-9
0-6 = Make (70%)
7-9 = Miss (30%)
One Trial = 2 digits
Both in {0-6}?
That's a success!
 Sample: 20 trials 4,2 1,5 3,8 0,6 9,2 5,1 7,4 6,0 ... 10/20
Result: 10 successes out of 20 = 50%. Theoretical: \( 0.7 \times 0.7 = 0.49 = 49\% \) ✓

More Trials = Better Estimate

0%
25%
50%
75%
Exp. Prob.
 10 50 100 500 1000 Number of Trials 49% Experimental Theoretical (49%)

Common Mistakes to Avoid

✕ Wrong Probability Match
Using a coin (50/50) to model a 40% event. The tool must match the event's probability.
✕ Too Few Trials
Running only 5 trials and trusting the result. Use at least 20-50 trials for a reasonable estimate.
✕ Wrong Trial Definition
Confusing "one trial" with "one step." A compound event trial includes all steps (e.g., both free throws).

Summary Cheat Sheet

Step What to Do Example
1. Identify State the compound event "Make both free throws"
2. Choose Tool Match probabilities Digits 0-6 = make (70%)
3. Define Trial Describe all steps + success 2 digits, both in {0-6}
4. Run Trials Repeat 20-50+ times 30 trials recorded in table
5. Analyze \(\frac{\text{successes}}{\text{trials}}\) \(\frac{15}{30} = 50\%\)

Question 1 of 5

A store gives away \(1\) of \(4\) toys with equal probability. You want to simulate buying bags until you get all 4 toys. Which tool best models this?

Question 2 of 5

A simulation uses random digits 0-9 to model a \(30\%\) probability. Which digit assignment is correct?

Question 3 of 5

In a free-throw simulation (\(70\%\) make rate), one trial consists of generating 2 random digits. In 25 trials, 12 had both digits in {0-6}. What is the experimental probability of making both shots?

Question 4 of 5

Why do simulations produce different results each time you run them?

Question 5 of 5

You flip a coin and roll a die. You want to find the probability of getting heads AND a number greater than \(4\). Which correctly defines one trial?