🔍 Session 2.5 - Conditional Probability – Visual and Numeric Example

What is Conditional Probability?

Sometimes probabilities need to be updated when we learn new information. Conditional probability answers this question:

“What is the probability of event B, given that event A has happened?”

This is written as P(B | A) and pronounced “the probability of B given A.”

Interpretation as Relative Frequency (Simple Case)

In a case where all outcomes of a random experiment are equally likely:

Then:

P(B | A) = P(A ∩ B) / P(A) = (number of outcomes in A ∩ B) / (number of outcomes in A)

This means conditional probability is like the relative frequency of B among all the cases where A happened.

Example 2.18 – Surface Flaws and Defective Parts

Total parts: 400

Flawed parts (F): 40
→ Defective (D ∩ F): 10
→ Not defective (D' ∩ F): 30

Non-flawed parts (F′): 360
→ Defective (D ∩ F′): 18
→ Not defective (D' ∩ F′): 342

🔢 Match the Calculations to the Formula

We want P(D | F):

P(D | F) = P(D ∩ F) / P(F) = 10 / 40 = 0.25

We want P(D | F′):

P(D | F′) = P(D ∩ F′) / P(F′) = 18 / 360 = 0.05

Interpretation: The probability of being defective is 5× higher if the part has a surface flaw. This is the power of conditional probability — adjusting based on known conditions.

🎓 Practice Question

There are 20 students in a class. 10 students submitted an assignment. Among them:

What is the probability that a student got an A, given that they submitted the assignment?

Solution:

We are asked for: P(A | Submitted)

Total who submitted = 10

Number who got A among those = 3

So,

P(A | Submitted) = 3 / 10 = 0.3