gen-define-conditional-probability-and-give-one-short-worked-exa-f9be

Define conditional probability and give one short worked example.

# Conditional Probability **Definition:** Conditional probability is the probability of an event A occurring given that another event B has already occurred. It's denoted as P(A|B) and calculated as: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$ where P(B) > 0. ## Worked Example **Problem:** A class has 30 students: 18 females and 12 males. There are 10 students who wear glasses: 7 females and 3 males. If we randomly select a student who is female, what's the probability she wears glasses? **Solution:** - Let A = "wears glasses" - Let B = "is female" We want P(A|B). $$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{7/30}{18/30} = \frac{7}{18}$$ **Answer:** The probability is 7/18 ≈ 0.389 or about 38.9% *Intuition:* Once we know the student is female, we only consider the 18 females, of which 7 wear glasses, giving us 7/18.