Tree diagrams help visualize sequences of events and calculate complex probabilities systematically.
Branches: Show possible outcomes
Probabilities: Written on branches
Endpoints: Show final outcomes
Multiplication: Along branches
P(A|B): Probability of A given B
Formula: P(A|B) = P(A and B) / P(B)
Example: P(Rain|Cloudy)
P(HH) = 0.5 × 0.5 = 0.25
P(exactly one H) = P(HT) + P(TH) = 0.25 + 0.25 = 0.5
P(at least one H) = 1 - P(TT) = 1 - 0.25 = 0.75
Along branches: Multiply probabilities
Different paths: Add probabilities
All outcomes sum to 1: 0.25 + 0.25 + 0.25 + 0.25 = 1
Scenario: A class has 30 students. 18 play football, 12 play basketball, 6 play both sports.
Total students: 30
Play football (F): 18
Play basketball (B): 12
Play both (F ∩ B): 6
P(B|F) = P(plays basketball given plays football)
= P(B ∩ F) / P(F)
= (6/30) / (18/30) = 6/18 = 1/3
Question 1: Drawing two cards without replacement from a deck. What is P(both are hearts)?
Hint: First card P(Heart) = 13/52, Second card P(Heart|First was Heart) = 12/51
Question 2: A bag has 4 red and 6 blue balls. Draw 2 balls without replacement. What is P(first red, second blue)?
Question 3: In a class, P(studies Math) = 0.8, P(passes test) = 0.9, P(passes test | studies Math) = 0.95. What is P(studies Math | passes test)?