Computing Probabilities and Simulation

Lesson 2 of 2

In this lesson:

• Compute compound event probabilities from complete sample spaces
• Design simulations to estimate compound probability
• Interpret simulation results as probability approximations

Lesson 2 Learning Objectives Today

By the end of this lesson, you should be able to:

4. Compute the probability of a compound event using P = favorable/total
5. Design and use a simulation to estimate the probability of a compound event
6. Interpret simulation results as an approximation of theoretical probability

From Sample Space to Probability

You built the two-dice table in Lesson 1 — 36 outcomes total.

The formula is the same as for simple events:

The complexity is in building the correct complete sample space — not in the formula.

Using the Table: P(sum = 7)

From the two-dice table (Lesson 1):

• Highlighted cells: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — 6 favorable

Using the Table to Find P(sum ≥ 10)

Count pairs where sum = 10, 11, or 12:

• Sum = 10: (4,6), (5,5), (6,4) → 3 pairs
• Sum = 11: (5,6), (6,5) → 2 pairs
• Sum = 12: (6,6) → 1 pair

P(at least one 3): Avoiding Double-Count

Die 1 = 3 OR Die 2 = 3 — but (3,3) is in both row 3 and column 3.

• Row 3: 6 outcomes; Column 3: 6 outcomes; overlap: (3,3) counted twice

Guided Practice: P(doubles) from Table

Using the two-dice table:

• Doubles = both dice show the same value
• Locate all cells where Die 1 = Die 2

Count the favorable outcomes and write .

P(doubles) Answer from the Table

Favorable: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) → 6 outcomes

Using the Tree: Three Coin Probabilities

From the 3-coin tree: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} — 8 paths.

Check-In: Coin Probability from Tree

From the 3-coin tree:

What is P(first coin = H AND third coin = T)?

List the favorable paths, then compute the probability.

Check-In Answer: First H and Third T

Favorable paths where flip 1 = H and flip 3 = T:

• HHT ✓ (H, H, T)
• HTT ✓ (H, T, T)

Count: 2 favorable paths

When Sample Spaces Are Too Large

The two-dice table works because 36 outcomes is manageable.

What if we needed: "Probability that at least one of 5 friends has a birthday on a weekend"?

• Each person: 7 possible birthday days → $7^5 = $ 16,807 outcomes

Enumerating is impractical. Simulation gives an approximation.

Simulation Design: Four Required Decisions

Every simulation needs four design decisions:

1. Random mechanism — what mimics the real probability?
2. One trial — what counts as one repetition?
3. Success — what outcome counts as the event?
4. Repetitions — how many trials to run?

Applying the Template: Weekend Birthday

Event: At least one of 5 friends has a birthday on a weekend.
per person.

Use random digits 1–7: {1, 2} = weekend; {3, 4, 5, 6, 7} = weekday.

Running the Simulation: Class Pooling

Simpler simulation: P(both flips are heads) = P(HH) — theoretical = 1/4 = 0.25

Mechanism: Roll a die — 1, 2, 3 = Heads; 4, 5, 6 = Tails. Two rolls = one trial.

Class run: Each student does 4 trials. Pool all results.

Compare class estimate to 0.25.

Practice: Design Your Own Simulation

3-child family. Estimate , P(girl) = 1/2.

1. Mechanism: Coin — Heads = girl, Tails = boy
2. Trial: Flip 3 times
3. Success: At least 2 heads
4. Repetitions: 30 trials

Record results and compute your estimate.

Compare Simulation to Theory: 3-Child Family

Theoretical calculation: List all outcomes from the 3-coin tree:

Outcomes with ≥ 2 girls: GGG, GGB, GBG, BGG → 4 out of 8

How close was your simulation estimate?

Lesson 2 Summary and Misconception Review

• — complete sample space required
• Simulation estimates probability when enumeration is impractical

Watch out:

• Stage order fixed; HT ≠ TH
• Total = ; count tree leaves not branches
• "At least N" includes N and beyond
• Simulation results vary — not an error

Compound Events: Complete Lesson Summary

This two-lesson unit covered:

• Defining compound events and the combined sample space
• Building sample spaces: organized lists, tables, tree diagrams
• Computing by counting and dividing
• Designing and interpreting simulations

Next: High school probability builds on this with counting principles and formal probability rules.