Link to my example of when i used to be a child and visual intuition vs the other way round.

Statistical Independence: Homework and Teacher Collection

Understanding Independence Through Examples

Three Scenarios of Dependence/Independence

Scenario 1: Perfect Positive Dependence

• Each time you don’t do homework, the teacher forgets to collect it
• The events guarantee each other - they are perfectly dependent
• If you know one event, you can predict the other with 100% certainty

Scenario 2: Perfect Negative Dependence

• When you don’t do homework, the teacher never forgets to collect it
• If you skip homework, probability teacher forgets = 0
• The events are strongly dependent (inversely related)

Scenario 3: Independence

• Whether you do homework has no effect on whether teacher collects it
• Even knowing you didn’t do homework, teacher still has same 50% chance of forgetting
• The events are independent

Mathematical Test for Independence

Two events A and B are independent if: P(A and B) = P(A) × P(B) (here I will give the other formula for independence test)

Observed Data: 10 Days of Homework(Here i will use balls and colors so with pictures)

Day	You Did Homework	Teacher Collected	Outcome
1	Yes	Yes	✓✓
2	Yes	No	✓✗
3	No	Yes	✗✓
4	Yes	Yes	✓✓
5	No	No	✗✗
6	Yes	No	✓✗
7	No	Yes	✗✓
8	Yes	Yes	✓✓
9	No	No	✗✗
10	Yes	Yes	✓✓

Data Summary for Independence

• You did homework: 6 out of 10 days → P(You did) = 0.6
• Teacher collected: 6 out of 10 days → P(Teacher collected) = 0.6
• Both happened: 4 out of 10 days → P(Both) = 0.4

Independence Test

For independence: P(Both) should equal P(You did) × P(Teacher collected)

• Expected if independent: 0.6 × 0.6 = 0.36
• Observed: 0.4
• Result: Close to independent (0.4 ≈ 0.36)

Making Data Perfectly Independent(Here I will introduce the other formula for indepdence pa times pb as a good way to solve for intersection that gives perfect independence)

To make the intersection exactly match independence, we need: P(Both) = 0.6 × 0.6 = 0.36

Since we have 10 days, we need: 0.36 × 10 = 3.6 ≈ 4 days where both events occur.

Our data already shows 4 days where both happened, so it’s essentially independent!

What Happens with Different Intersections(Here I will calls the intuitive sense of the reader to say that more means more )

If They Share Too Much (High Positive Dependence)

• Intersection = 6 days (instead of 4)
• P(Both) = 0.6 > 0.36 expected
• When you do homework, teacher is more likely to collect
• Strong positive dependence

If They Share Too Little (Negative Dependence)

• Intersection = 1 day (instead of 4)
• P(Both) = 0.1 < 0.36 expected
• When you do homework, teacher is less likely to collect
• Negative dependence (events avoid each other)