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

Larson/Farber Ch. 3 Check in assignment 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)

Larson/Farber Ch. 3 Check in assignment Independent 6. Dependent 7. Dependent 8. Independent 9a b c. Dependent 10a b c. Dependent 11a b a b a b c a b c d. Dependent

Larson/Farber Ch. 3 Check in assignment a b c d. Dependent 17a b c a b c a b c d a b c a b a b a b c. Yes 25d. Answers will vary

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: #1-18 all

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

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.

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.

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

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.

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

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 – 2/52 = 28/52 = 0.538

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 = When events are mutually exclusive, P(A or B) = P(A) + P(B)

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 Yes No Undecided Total One of the responses is selected at random. Find : 1. P(Miami and Yes) 2. P(Miami and Seattle)

Larson/Farber Ch. 3 Contingency Table 1. P(Miami and Yes) 2. P(Miami and Seattle) = 250/ /250 = 150/1000 = 0.15 = 0= 0 OmahaSeattleMiamiTotal Yes No Undecided Total One of the responses is selected at random. Find:

Larson/Farber Ch. 3 Contingency Table 3 P(Miami or Yes) 4. P(Miami or Seattle) 250/ /1000 – 0/1000 = 700/1000 = 0.7 OmahaSeattleMiamiTotal Yes No Undecided Total / /1000 – 150/1000 = 500/1000 = 0.5

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