🧠 Session 2.6- Intersections of Events and Multiplication and Total Probability Rule

Multiplication Rule: Intersections of Events

When we want to know the chance that both events A and B happen, we use:

P(A ∩ B) = P(B | A) × P(A) = P(A | B) × P(B)

This tells us how to find the chance that two events overlap. It’s especially useful when we know a conditional probability.

Example 1: Machining Stages

Let A = first stage meets specs (P(A) = 0.90)
B = second stage meets specs given A (P(B|A) = 0.95)

P(A ∩ B) = P(B|A) × P(A) = 0.95 × 0.90 = 0.855

Interpretation: There’s an 85.5% chance that both stages meet specifications. The more steps, the lower the chance unless each step is very reliable!

Example 2: Student Assignment

Out of 20 students, 10 submitted the homework. Among them, 3 got an A. What is P(A | submitted)?

P(A | submitted) = Number of A's / Number who submitted = 3 / 10 = 0.30
P(A ∩ submitted) = P(A | submitted) × P(submitted) = 0.30 × (10/20) = 0.15

Interpretation: Given that a student submitted, 30% of them earned an A. Looking at the whole class, 15% of all students both submitted and earned an A.

Example 3: Package Delivery

60% of packages are sent via Air (A), and 90% of those arrive on time (T). Find P(package is Air and On Time).

P(A ∩ T) = P(T | A) × P(A) = 0.90 × 0.60 = 0.54

Interpretation: More than half of all packages are both sent by Air and arrive on time.

Total Probability Rule

If event B can happen under multiple conditions, we combine them using:

P(B) = P(B | A) × P(A) + P(B | A′) × P(A′)

Example 4: Semiconductor Contamination

20% of chips are exposed to high contamination (H).
If contaminated: P(Failure | H) = 0.10
If clean: P(Failure | H′) = 0.005

P(Failure) = 0.10 × 0.20 + 0.005 × 0.80 = 0.024

Interpretation: The total failure rate is a weighted average based on contamination level.

FYI for Advanced Students:

More generally, if events E₁, E₂, ..., Eₖ are mutually exclusive and exhaustive:

P(B) = Σ [P(B | Eᵢ) × P(Eᵢ)] for i = 1 to k

This breaks down B into smaller parts that can be calculated separately and summed.

📝 Practice Question

Out of 30 students, 12 completed the project. Among those, 5 received an A grade. What is:

P(A | completed) = 5 / 12 = 0.4167

P(completed) = 12 / 30 = 0.40

P(A ∩ completed) = 0.4167 × 0.40 = 0.1667

Interpretation: Among the students who completed the project, about 41.7% got an A. Looking at the entire class of 30, about 16.7% of all students both completed the project and earned an A.