Overview
This lecture explains conditional probability, the key formula used to compute it, and demonstrates its application through several probability questions involving dice, students, and birds using contingency tables.
Introduction to Conditional Probability
- Conditional probability measures the likelihood of an event A occurring given that event B has already occurred.
- It is notated as P(A|B), which means "probability of A given B".
- The core question is: How much of A is contained within B?
Conditional Probability Formula
- The formula is P(A|B) = P(A and B) / P(B).
- P(A and B) is the probability both A and B happen simultaneously.
- P(B) is the probability that B occurs.
Dice Example
- Sample space for a six-sided die: {1, 2, 3, 4, 5, 6}.
- Let A = {1, 3, 5} (odd numbers), B = {3, 4, 5}.
- Elements in both A and B: {3, 5} (2 elements).
- P(A|B) = 2/3 since 2 of the 3 elements in B are also in A.
- Using the formula: P(A and B) = 2/6, P(B) = 3/6, so P(A|B) = (2/6) / (3/6) = 2/3.
Student Enrollment Example
- 500 students: 150 in algebra, 80 in chemistry, 30 in both.
- Probability a student is taking algebra: 150/500 = 3/10 or 30%.
- Probability of chemistry given algebra: 30/150 = 1/5 or 20%.
- Probability of algebra given chemistry: 30/80 = 3/8 or 37.5%.
Birds in a Zoo Example (Contingency Table)
- 200 birds: 70 male brown eyes, 100 female brown eyes, 20 male blue eyes, 10 female blue eyes.
- Total males: 90; total females: 110; brown-eyed: 170; blue-eyed: 30.
- Probability bird is female: 110/200 = 11/20 or 55%.
- Probability male with brown eyes: 70/200 = 7/20 or 35%.
- Probability female given brown eyes: 100/170 = 10/17 ≈ 58.8%.
- Probability male given blue eyes: 20/30 = 2/3 or 66.7%.
- Probability blue eyes given female: 10/110 = 1/11 ≈ 9.1%.
Key Terms & Definitions
- Conditional Probability (P(A|B)) — Probability of event A occurring given event B has occurred.
- Intersection (A and B) — Outcomes where both A and B happen.
- Contingency Table — Table summarizing the frequency of events with two or more variables.
Action Items / Next Steps
- Practice using the conditional probability formula on similar problems.
- Review construction and analysis of contingency tables.
- Complete any assigned textbook exercises on conditional probability.