1. Larson/Farber Ch. 3 Section 3.3 The Addition Rule Statistics Mrs. Spitz Fall 2008

2. Larson/Farber Ch. 3 Check in assignment 3.2 1.Two events are independent if the occurrence of one of the events does not affect the probability of the occurrence of the other event. If P(B|A) = P(B) or P(A|B) = P(A), then Events A and B are independent. 2a. Roll a die twice. The outcome of the 2 nd toss is independent of the outcome of the 1 st toss. 2b. Draw two cards(without replacement) from a standard 52 card deck. The outcome of the 2 nd card is dependent upon the outcome of the 1 st card. 3. False. If two events are independent, P(A|B) = P(A) 4. False. If events A and B are independent, then P(A and B) = P(A)●P(B)

3. Larson/Farber Ch. 3 Check in assignment 3.2 5. Independent 6. Dependent 7. Dependent 8. Independent 9a. 0.8 9b. 0.0032 9c. Dependent 10a. 0.740 10b. 0.822 10c. Dependent 11a. 0.0168 11b. 0.93 12a. 0.24 12b. 0.6 13a. 0.109 13b. 0.382 13c. 0.618 14. 0.083 15a. 0.839 15b. 0.167 15c. 0.506 15d. Dependent

4. Larson/Farber Ch. 3 Check in assignment 3.2 16a. 0.556 16b. 0.525 16c. 0.167 16d. Dependent 17a. 0.0000000243 17b. 0.859 17c. 0.141 18a. 0.055 18b. 0.238 18c. 0.762 19a. 0.2 19b. 0.04 19c. 0.008 19d. 0.488 20a. 0.985 20b. 0.015 20c. 0.000000125 21. 0.954 22. 0.933 23a. 0.444 23b. 0.4 24a. 0.074 24b. 0.999 25a. 0.462 25b. 0.538 25c. Yes 25d. Answers will vary

5. Larson/Farber Ch. 3 Objectives/Assignment How to determine if two events are mutually exclusive How to use the addition rule to find the probability of two events. Assignment: 129-131 #1-18 all

6. Larson/Farber Ch. 3 What is different? In probability and statistics, the word “or” is usually used as an “inclusive or” rather than an “exclusive or.” For instance, there are three ways for “Event A or B” to occur. –A occurs and B does not occur –B occurs and A does not occur –A and B both occur

7. Larson/Farber Ch. 3 Independent does not mean mutually exclusive Students often confuse the concept of independent events with the concept of mutually exclusive events.

8. Larson/Farber Ch. 3 Study Tip By subtracting P(A and B), you avoid double counting the probability of outcomes that occur in both A and B.

9. Larson/Farber Ch. 3 Compare “A and B” to “A or B” The compound event “A and B” means that A and B both occur in the same trial. Use the multiplication rule to find P(A and B). The compound event “A or B” means either A can occur without B, B can occur without A or both A and B can occur. Use the addition rule to find P(A or B). A B A or B A and B A B

10. Larson/Farber Ch. 3 Mutually Exclusive Events Two events, A and B, are mutually exclusive if they cannot occur in the same trial. A = A person is under 21 years old B = A person is running for the U.S. Senate A = A person was born in Philadelphia B = A person was born in Houston A B Mutually exclusive P(A and B) = 0 When event A occurs it excludes event B in the same trial.

11. Larson/Farber Ch. 3 Non-Mutually Exclusive Events If two events can occur in the same trial, they are non-mutually exclusive. A = A person is under 25 years old B = A person is a lawyer A = A person was born in Philadelphia B = A person watches West Wing on TV A B Non-mutually exclusive P(A and B) ≠ 0 A and B

12. Larson/Farber Ch. 3 The Addition Rule The probability that one or the other of two events will occur is: P(A) + P(B) – P(A and B) A card is drawn from a deck. Find the probability it is a king or it is red. A = the card is a king B = the card is red. P(A) = 4/52 and P(B) = 26/52 but P(A and B) = 2/52 P(A or B) = 4/52 + 26/52 – 2/52 = 28/52 = 0.538

13. Larson/Farber Ch. 3 The Addition Rule A card is drawn from a deck. Find the probability the card is a king or a 10. A = the card is a king B = the card is a 10. P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52 P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054 When events are mutually exclusive, P(A or B) = P(A) + P(B)

14. Larson/Farber Ch. 3 The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is: Contingency Table 3. P(Miami or Yes) 4. P(Miami or Seattle) OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 One of the responses is selected at random. Find : 1. P(Miami and Yes) 2. P(Miami and Seattle)

15. Larson/Farber Ch. 3 Contingency Table 1. P(Miami and Yes) 2. P(Miami and Seattle) = 250/1000 150/250 = 150/1000 = 0.15 = 0= 0 OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 One of the responses is selected at random. Find:

16. Larson/Farber Ch. 3 Contingency Table 3 P(Miami or Yes) 4. P(Miami or Seattle) 250/1000 + 450/1000 – 0/1000 = 700/1000 = 0.7 OmahaSeattleMiamiTotal Yes100150 400 No12513095 350 Undecided 75170 5 250 Total3004502501000 250/1000 + 400/1000 – 150/1000 = 500/1000 = 0.5

17. Larson/Farber Ch. 3 Probability at least one of two events occur P(A or B) = P(A) + P(B) - P(A and B) Add the simple probabilities, but to prevent double counting, don’t forget to subtract the probability of both occurring. For complementary events P(E') = 1 - P(E) Subtract the probability of the event from one. The probability both of two events occur P(A and B) = P(A) P(B|A) Multiply the probability of the first event by the conditional probability the second event occurs, given the first occurred. Summary