AP Statistics – Unit 4 Lesson 5: Conditional Probability

Topic 4.5 — Understanding “Given That…”

---

1. Warm-Up Questions

Write short answers.

1. If a student is known to be in 10th grade, how does that information change the probability that the student is taking AP Statistics? Explain your thinking.

2. Does learning new information usually change probabilities? Give one example of a time when it would.

These questions introduce the meaning of conditioning: probabilities change when new information is given.

---

2. Key Vocabulary & Concepts

Event A

Something that might happen.
Example: “Person has the disease.”

Event B

A condition or given information.
Example: “Person tested positive.”

Conditional Probability: P(A | B)

Pronounced: “Probability of A given B.”

This is the probability of A after we know B has occurred.

---

3. Conditional Probability Formula (UNC-2.D.1)

Formula:

P(A∣B)=P(B)P(A and B)​

Meaning:

Out of all cases where B happens, what proportion also has A?

Simple Example:

• 30% of students play sports (B)

• 12% play sports and play violin (A and B)

P(A∣B)=0.300.12​=0.40

Interpretation:
Among the students who play sports, 40% also play violin.

---

4. Using Two-Way Tables (UNC-2.D.2)

Example Table: Students & Part-Time Jobs

Sample Conditional Probabilities

1. P(Has Job | Sophomore)
Out of sophomores (200), 60 have jobs:

200/60​=0.30

2. P(Sophomore | Has Job)
Out of everyone with jobs (100), 60 are sophomores:

100/60​=0.60

Key idea: Direction matters.
P(A|B) is not the same as P(B|A).

---

5. Exploration: How Conditioning Changes Probabilities (UNC-2.D.3)

Example:

Rolling a die.

• P(rolling a 6) = 1/6

• But if you know the roll was even, possibilities shrink to {2, 4, 6}.
  So:

  P(6∣even)=31​

Conditioning focuses on a smaller group, which changes probabilities.

---

6. Think–Pair–Share Activity: “Positive Test Result”

Scenario: Disease Testing

A disease affects 2% of the population.

A test gives:

• True positive rate: 90%

• False positive rate: 8%

Meaning:

• P(test + | disease) = 0.90

• P(test + | no disease) = 0.08

Assume disease status and test accuracy are independent.

Tasks

1. Build a two-way table using 1000 people.

2. Fill in counts for:

   • True positives

   • False positives

   • True negatives

   • False negatives

3. Compute:

   • P(disease | test +)

   • P(no disease | test +)

4. With your partner, discuss:
   Why is the probability of truly having the disease much lower than most people expect after a positive test?

5. Share your interpretation with the class.

This builds real-world statistical literacy.

---

7. Main Activity – “Café Loyalty Program”

(Unique activity tied to conditional probability.)

A café tracks 600 customers:

• 240 are Loyalty Members (Event L)

• 360 are Non-Members

• 180 Members buy pastries (Event P)

• 90 Non-Members buy pastries

Fill in the table:

Activity Tasks

1. Complete the missing values.

2. Calculate:

   • P(P∣L) = probability a customer buys pastries given they are a member.

   • P(P∣Non-Member).

   • P(L∣P) = probability a pastry-buyer is a member.

3. Interpret each result in a sentence.

4. Explain whether membership seems to increase pastry buying.

5. Create one extra conditional probability question and solve it.

---

8. Real-World Interpretation Practice (Skill 4.B)

Answer in 1–2 sentences.

A. Why is understanding conditional probability essential for interpreting medical tests, weather forecasts, or school data?

B. Give an example where P(A|B) is very different from P(B|A). Explain in context.

---

9. Exit Question

In your own words, explain what conditional probability tells us and why it is important.