Discrete Math Section 16.2 Find the probability of events occurring together. Determine whether two events are independent. A sack contains 2 yellow and 3 green marbles. Two marbles are selected. What is the probability they are both the same color? The answer depends on whether the 1 st marble is replaced before the 2 nd is selected. P(YY) = 2/5 ∙ 2/5 = 4/25 P(YG) = 2/5 ∙ 3/5 = 6/25 P(GY) = 3/5 ∙ 2/5 = 6/25 P(GG) = 3/5 ∙ 3/5 = 9/25 Same color P(YY) or P(GG)= 4/25 + 9/25 = 13/25

No replacement P(YY) = 2/5 ∙ 1/4 = 2/20 P(YG) = 2/5 ∙ 3/4 = 6/20 P(GY) = 3/5 ∙ 2/4 = 6/20 P(GG) = 3/5 ∙ 2/4 = 6/20 Same color P(YY) or P(GG)= 2/20 + 6/20 = 2/5

In the 1 st example, probability that the 2 nd marble is green does not depend on the color of the 1 st ball. The 1 st ball is green and the 2 nd ball is green are independent in case 1 not so in case 2. Two events A and B are independent iff the occurrence of A does not affect the probability of B. P(B|A) = P(B) Conditional Probability P(B|A) means the probability of B after A has occurred.

Probability of Events Occurring Together Rule 1: For any two events A and B: P(A and B) = P(A) ∙ P(B|A) Rule 2: If events A and B are independent then: P(A and B) = P(A) ∙ P(B) From rule 1: P(B|A) = P(A and B) P(A) And P(B|A) = n(A П B) n(A)

example A card is randomly drawn from a standard deck of 52 cards. Are the events “jack” and “spade” independent? Question: Is P(S|J) = P(S) ? Note: P(S|J) = n(JП S) n(J)

example Each student in a class of 30 students studies one foreign language and one science class. a. Find the probability that a randomly chosen student studies chemistry. b. Find the probability that a randomly chosen student studies chemistry given that the student studies French. c. Are the events “chemistry” and “French” independent?

Assignment Page 609 Problems 1,6,8,10,13,14,16,17,18,22,26,27