Addition and Multiplication Rules of Probability Section 4.4

Objectives Understand the difference between addition and multiplication rules. Apply the addition and multiplication rules of probability.

P(A or B) = P(A) + P(B) – P(A ∩ B) General Addition Rule P(A or B) = P(A) + P(B) – P(A ∩ B) When to Apply the Addition Rule Any outcome in A Any outcome in B Any outcome in Both A and B

Example-Addition Rule An ice cream cup has 3 scoops of chocolate, 4 scoops of vanilla, and 3 scoops of butter scotch ice cream. What is the probability that if you randomly sampled the ice cream you will taste chocolate or vanilla? Solution: P(C ∪ V) = P(C) + P(V) – P(C ∩ V) P(C ∪ V) = 3/10 + 4/10 - 0 P(C ∪ V) = 7/10

Example-Addition Rule A single standard die is rolled. What is the probability of rolling a six or a one? Solution: P(6 ∪ 1) = P(6) + P(1) – P(6 ∩ 1) P(6 ∪ 1) = 1/6 + 1/6 – 0 P(6 ∪ 1) = 2/6 P(6 ∪ 1) = 1/3

Example-Addition Rule Robert is playing cards. What is the probability of pulling a king or a jack? Solution: P(K ∪ J) = P(K) + P(J) – P(K ∩ J) P(K ∪ J) = 4/52 + 4/52 – 0 P(K ∪ J) = 8/52 P(K ∪ J) = 2/13

Example-Addition Rule Mary is playing cards. What is the probability of randomly pulling a heart or a club out of the deck? Solution: P(H ∪ C) = P(H) + P(C) – P(H ∩ J) P(H ∪ C) = 13/52 + 13/52 – 0 P(H ∪ C) = 26/52 P(H ∪ C) = 1/2

General Multiplication Rule P(A and B) = P(A) x P(B) When to Apply the Multiplication Rule To find the intersection of A and B If dependent, P(B) becomes P(B | A) This is read, Probability of B given A has already happened

Example-Multiplication Rule Christy has a bag of candies. In the bag there are 5 red colors, 3 orange colors and 8 green colors. She takes one candy, records its color and puts it back in the bag. She then draws another candy. What is the probability of taking out a green candy followed by a red candy? Solution: P(G ∩ R) = P(G) * P(R, given G) P(G ∩ R) = 8/16 * 5/16 P(G ∩ R) = 40/256 P(G ∩ R) = 5/32

Example-Multiplication Rule Dena has a box with 7 blue marbles and 3 pink marbles. Two marbles are drawn without replacement from the box. What is the probability that both of the marbles are blue? Solution: P(B ∩ B) = P(B) * P(B | B) P(B ∩ B) = (7/10) * (6/9) P(B ∩ B) = 42/90 P(B ∩ B) = 7/15

Example-Multiplication Rule John is going to draw two cards from standard deck. What is the probability that the first card is a queen and the second card is a jack? Solution: P(Q ∩ J) = P(Q) * P(J | Q) P(Q ∩ J) = (4/52) * (4/51) P(Q ∩ J) = 16/2652 P(Q ∩ J) = 4/663 or 0.006

Example-Multiplication Rule Emily has a folder of colored paper. The folder has 4 yellow color papers, 5 black color papers, 6 white color papers, and 5 brown color papers. She picks one paper, record its color, puts it back in the folder and draws another paper. What is the probability of taking out a brown paper followed by a yellow paper? Solution: P(B ∩ Y) = P(B) * P(Y | B) P(B ∩ Y) = (4/10) * (4/9) P(B ∩ Y) = 20/400 P(B ∩ Y) = 1/20

4.4 Addition and Multiplication Rules of Probability Summarize Notes Read section 4.4 Homework Worksheet