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1 Section 12.5 Tree Diagrams

2 What You Will Learn Counting Principle Tree Diagrams

3 Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M N distinct ways

4 Definitions Sample space: A list of all possible outcomes of an experiment. Sample point: Each individual outcome in the sample space. Tree diagrams are helpful in determining sample spaces

5 Example 1: Selecting Balls without Replacement Two balls are to be selected without replacement from a bag that contains one red, one blue, one green and one orange ball. a) Use the counting principle to determine the number of points in the sample space. b) Construct a tree diagram and list the sample space. c) Find the probability that one orange ball is selected. d) Find the probability that a green ball followed by a red ball is selected

6 12.5-8

7 Example 3: Selecting Ticket Winners A radio station has two tickets to give away to a Bon Jovi concert. It held a contest and narrowed the possible recipients down to four people: Christine (C), Mike Hammer (MH), Mike Levine (ML), and Phyllis (P). The names of two of these four people will be selected at random from a hat, and the two people selected will be awarded the tickets. a) Use the counting principle to determine the number of points in the sample space. b) Construct a tree diagram and list the sample space. c) Determine the probability that Christine is selected. d) Determine the probability that neither Mike Hammer nor Mike Levine is selected. e) Determine the probability that at least one Mike is selected

8 b) Construct a tree diagram and list the sample space

9 P(event happening at least once) P event happening at least once event does 1 P not happen

10 Section 12.6 OR and AND Problems

11 What You Will Learn Compound Probability OR Problems AND Problems Independent Events

12 Compound Probability In this section, we learn how to solve compound probability problems that contain the words and or or without constructing a sample space

13 OR Probability The or probability problem requires obtaining a successful outcome for at least one of the given events

14 Probability of A or B To determine the probability of A or B, use the following formula. P(A or B) P(A) P(B) P(A and B)

15 Example 1: Using the Addition Formula Each of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 is written on a separate piece of paper. The 10 pieces of paper are then placed in a hat, and one piece is randomly selected. Determine the probability that the piece of paper selected contains an even number or a number greater than

16 Mutually Exclusive Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously. If two events are mutually exclusive, then the P(A and B) = 0. The addition formula simplifies to P(A or B) P(A) P(B)

17 Example 3: Probability of A or B One card is selected from a standard deck of playing cards. Determine whether the following pairs of events are mutually exclusive and determine P (A or B). a) A = an ace, B = a 9 b) A = an ace, B = a heart c) A = a red card, B = a black card d) A = a picture card, B = a red card

18 And Problems The and probability problem requires obtaining a favorable outcome in each of the given events

19 Probability of A and B To determine the probability of A and B, use the following formula. P(A and B) P(A) P(B)

20 Probability of A and B Since we multiply to find P (A and B), this formula is sometimes referred to as the multiplication formula. When using the multiplication formula, we always assume that event A has occurred when calculating P(B) because we are determining the probability of obtaining a favorable outcome in both of the given events

21 Example 5: An Experiment without Replacement Two cards are to be selected without replacement from a deck of cards. Determine the probability that two spades will be selected

22 Independent Events Event A and event B are independent events if the occurrence of either event in no way affects the probability of occurrence of the other event. Rolling dice and tossing coins are examples of independent events

23 Example 6: Independent or Dependent Events? One hundred people attended a charity benefit to raise money for cancer research. Three people in attendance will be selected at random without replacement, and each will be awarded one door prize. Are the events of selecting the three people who will be awarded the door prize independent or dependent events?

24 Independent or Dependent Events? In general, in any experiment in which two or more items are selected without replacement, the events will be dependent

25 Section 12.7 Conditional Probability

26 What You Will Learn Conditional Probability

27 Conditional Probability In general, the probability of event E 2 occurring, given that an event E 1 has happened (or will happen; the time relationship does not matter), is called a conditional probability and is written P(E 2 E 1 )

28 Example 1: Using Conditional Probability A single card is selected from a deck of cards. Determine the probability it is a club, given that it is black

29 Conditional Probability For any two events E 1 and E 2, the conditional probability, P(E 2 E 1 ), is determined as follows. n E and E 1 2 P E 2 E 1 n E

30 Example 3: Using the Conditional Probability Formula Two hundred and fifty patients who had knee, hip, or heart surgery were asked whether they were satisfied, dissatisfied, or neutral regarding the results of their surgery. The responses are given in the table on the next slide

31 Example 3: Using the Conditional Probability Formula If one person from the 250 patients surveyed is selected at random, determine the probability that the person

32 Example 3: Using the Conditional Probability Formula a) was satisfied with the results of the surgery

33 Example 3: Using the Conditional Probability Formula b) was satisfied with the results of the surgery, given that the person had knee surgery

34 Example 3: Using the Conditional Probability Formula c) was dissatisfied with the results of the surgery, given that the person had hip surgery

35 Example 3: Using the Conditional Probability Formula d) had heart surgery, given that the person was dissatisfied with the results of the surgery

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