Discrete Math Section 16.2Find 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 1st marble is replaced before the 2nd is selected.

P(YY) = 2/5 2/5 = ∙ 4/25P(YG) = 2/5 3/5 = ∙ 6/25P(GY) = 3/5 2/5 = ∙ 6/25P(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 = ∙ 2/20P(YG) = 2/5 3/4 = ∙ 6/20P(GY) = 3/5 2/4 = ∙ 6/20P(GG) = 3/5 2/4 = ∙ 6/20

Same color P(YY) or P(GG)= 2/20 + 6/20 = 2/5

In the 1st example, probability that the 2nd marble is green does not depend on the color of the 1st ball. The

1st ball is green and the 2nd 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 TogetherRule 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