Lesson 3: Conditional Probability & Independence

Real-World Example:

Scenario: What's the probability a student passes a statistics exam?

• Without information: P(pass) might be 0.75
• Given they studied 10+ hours: P(pass | studied) might be 0.92
• Given they didn't study: P(pass | didn't study) might be 0.40

The additional information (whether they studied) changes the probability!

Example 1: Drawing Cards

Question: You draw one card from a standard deck. Given that the card is red, what is the probability it's a heart?

Solution:

1. Let A = "card is a heart"
2. Let B = "card is red"
3. We want P(A|B) = P(heart | red)
4. There are 26 red cards total (13 hearts + 13 diamonds)
5. Of these 26 red cards, 13 are hearts
6. P(heart | red) = 13/26 = 1/2

Alternative approach using formula:

• P(heart and red) = P(heart) = 13/52 = 1/4 (all hearts are red)
• P(red) = 26/52 = 1/2
• P(heart | red) = (1/4) / (1/2) = 1/2

Example 2: Rolling Dice

Question: You roll a fair die. Given that you rolled an even number, what's the probability you rolled a number greater than 3?

Solution:

1. Let A = "number > 3" = {4, 5, 6}
2. Let B = "even number" = {2, 4, 6}
3. A and B = {4, 6} (even numbers greater than 3)
4. Given B occurred, our restricted sample space is {2, 4, 6}
5. Of these 3 outcomes, 2 satisfy A (the 4 and 6)
6. P(A|B) = 2/3

Using formula:

• P(A and B) = 2/6 = 1/3
• P(B) = 3/6 = 1/2
• P(A|B) = (1/3) / (1/2) = 2/3

Example 3: Student Survey

A survey of 200 students asked about their year (Freshman/Sophomore) and whether they have a job.

	Has Job	No Job	Total
Freshman	30	70	100
Sophomore	60	40	100
Total	90	110	200

Question 1: What is P(has job)?

Solution: P(has job) = 90/200 = 0.45 or 45%

Question 2: What is P(has job | Freshman)?

Solution:

• Given: Student is a Freshman (restrict to that row)
• Total Freshmen: 100
• Freshmen with jobs: 30
• P(has job | Freshman) = 30/100 = 0.30 or 30%

Question 3: What is P(has job | Sophomore)?

Solution:

• Given: Student is a Sophomore (restrict to that row)
• Total Sophomores: 100
• Sophomores with jobs: 60
• P(has job | Sophomore) = 60/100 = 0.60 or 60%

Observation: Sophomores are twice as likely to have a job as Freshmen! (60% vs 30%)

Example 4: Drawing Without Replacement

Question: You draw 2 cards from a deck without replacement. What's the probability both are aces?

Solution:

1. Let A₁ = "first card is an ace"
2. Let A₂ = "second card is an ace"
3. We want P(A₁ and A₂)
4. P(A₁) = 4/52 (4 aces in 52 cards)
5. P(A₂|A₁) = 3/51 (after drawing one ace, 3 aces left in 51 cards)
6. P(A₁ and A₂) = (4/52) × (3/51) = 12/2652 = 1/221 ≈ 0.0045 or 0.45%

Notice: The second probability depends on what happened first (without replacement = dependent events)

Example 5: Multiple Events

Question: A box contains 5 red and 3 blue marbles. You draw 3 marbles without replacement. What's the probability all 3 are red?

Solution:

1. P(R₁) = 5/8 (first is red)
2. P(R₂|R₁) = 4/7 (second is red, given first was red)
3. P(R₃|R₁ and R₂) = 3/6 = 1/2 (third is red, given first two were red)
4. P(all 3 red) = (5/8) × (4/7) × (1/2) = 20/112 = 5/28 ≈ 0.179 or 17.9%

Example 6: Flipping Coins (Independent)

Question: You flip a fair coin twice. What's the probability of getting heads both times?

Solution:

The flips are independent (first flip doesn't affect second)

• P(H₁) = 1/2
• P(H₂) = 1/2
• P(H₁ and H₂) = (1/2) × (1/2) = 1/4 or 0.25

Example 7: Testing Independence with Data

Using our student survey from earlier:

	Has Job	No Job	Total
Freshman	30	70	100
Sophomore	60	40	100
Total	90	110	200

Question: Are "Year" and "Has Job" independent?

Solution:

Check if P(has job | Freshman) = P(has job)

• P(has job) = 90/200 = 0.45
• P(has job | Freshman) = 30/100 = 0.30
• 0.30 ≠ 0.45, so events are NOT independent

Conclusion: Year in school and having a job are dependent - being a sophomore makes you more likely to have a job!