2Tree DiagramUsed to show all of the possible outcomes of an experiment
3ExampleA couple plans on having 3 children. Assuming that the births are single births, make a tree diagram.
4SolutionThere are 8 birth orders.BBBBBGBGBBGGGBBGBGGGBGGG
5Questions Find each probability: All 3 children are girls There is one girlThere is at least one girlThere is at most one girl
6Answers Find each probability: All 3 children are girls 1/8 There is one girl 3/8There is at least one girl 7/8There is at most one girl 4/8
7Independent eventsTwo events, A and B, are independent if the occurrence of one event does not affect the probability of the occurrence of the other.
8Examples Rolling a pair of dice Tossing 2 coins Drawing 2 cards from a deck if the first card is replaced before the second card is drawn
9Dependent eventsTwo events, A and B, are dependent if the occurrence of one event does affect the probability of the occurrence of the other.
10ExamplesDrawing 2 cards from a deck of cards if the first card is not replaced before drawing the second card.Note: Without replacement is a clue that the events will be dependent.
11Multiplication Rule Independent Events P(A and B) = P(A) * P(B) P(A and B) = P(A)*P(B|A)P(B|A) means probability of B assuming that A has happened. It is called a conditional probability.
12ExampleA die is rolled twice. What is the probability that the first roll is an even number and the second roll is a number greater than 4?
13Solution These are independent events. P(even number) = 3/6 P(number > 4) = 2/6P(A and B) = 3/6 * 2/6= 6/36= 1/6
14ExampleTwo cards are drawn from a deck of cards. What is the probability that both are Kings, ifa. The first card is replaced before drawing the second cardb. The first card is not replaced before drawing the second card
15Solution There are 4 Kings in a deck of 52 cards. With replacement: = 0.006Without replacementP = 4/52 * 3 / 52= 0.005
16Tables to find conditional probabilities A sample of 1000 people was obtained. There were 500 men and 500 women. Of the men, 63 were left handed. Of the women, 50 were left handed.MenWomenTotalLeft handed6350113Right handed4374508875001000
17ExampleWhat is the probability that the person is a male given the person is right handed?
18ExampleWhat is the probability that the person is a male given the person is right handed?Solution: There were 887 right handed people. Of these, 437 were men.P(M|RH) = 437/887= 0.493
19ExampleWhat is the probability that person is right handed, given the person is male?
20ExampleWhat is the probability that person is right handed, given the person is male?Solution: There were 500 males. Of these, 437 were right handed.P(RH|M) = 437/500= 0.874
21Testing independence for a table Two events will be independent ifP(B|A) = P(B)
22ExampleAre the events “male” and “right handed” independent or dependent?
23Solution P(male) = 0.500 P(male|right handed) = 0.493 These are not equal, so the 2 events are dependent.Note: You could also see if P(right handed) = P(right handed|male)