Probability: Summary of Rules Rule 1: Complement Rule P(A) + P(Ac) = 1 if Ac represents the complement of A Rule 2B: Additive Rule P(A or B) = P(A) + P(B) if Events A and B are mutually exclusive Note: two events that are complements are always mutually exclusive Rule 3B: Multiplication Rule P(A and B) = P(A)×P(B) if Events A and B are independent

Clicker Quiz #8 Maria wants to take French or Spanish, or both. But classes are closed, ands he must apply to enroll in a language class. She has a 60% chance of being admitted to French, a 50% chance of being admitted to Spanish, and a 20% chance of being admitted to both French and Spanish. P(French or Spanish) = _____________________ = _______ = _____ P(French) +P(Spanish) – P(both) 0.6 + 0.5 – 0.2 0.9 The events “admitted to French” and “admitted to Spanish” are: Mutually exclusive but not independent Independent but not mutually exclusive Both mutually exclusive and independent Neither mutually exclusive nor independent

With 50 people in the room… There are 49 possible pairs with me. P(No match with my birthday) = (364/365)49 = 0.874 There are 49+48+47+…+1 = 1225 total possible pairs. P(No match at all) = .030 and we can estimate by (364/365)1225 = .035

Probability review Suppose we roll two fair dice, one red and one blue. Let the event A be that we roll the same number on both dice. Let the event B be that the sum of the two dice is even. What is P(A)? What is P(B)? What is P(A and B)? What is P(A or B)? Ans: 6/36 or 1/6 Ans: 18/36 or 1/2 Ans: 1/6 (Note: NOT independent) Ans: 1/2 (Note: NOT mutually exclusive)

More probability review Suppose that you roll a fair 6-sided die until the first occurrence of a 4. What is the probability that the first 4 occurs on the third roll? 1/6 5/6 (5/6) × (5/6) × (1/6) (1/6) × (1/6) × (1/6) (1/6) + (1/6) + (1/6)