Chapter 4

1. Chapter 4 Lecture 3 Sections: 4.4 – 4.5

2. Multiplication Rule Recall that we used addition for the P(A or B). the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once. We will now focus on finding the P(A and B). The probability that event A occurs in the first trial and the event that B occurs in the second trial. P(A and B): P(event A occurs in the first trial and the event that B occurs in the second trial) The word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B), but be sure that the probability of event B takes into account the previous occurrence of event A. Example 1. Say you flip a coin twice. What is the probability that you will get tails both times.

3. Tree Diagrams 2. What is the probability of guessing both questions correct on a 2 question quiz? The quiz consists a true/false question and a multiple choice question with possible answers (A, B, C). The figure below summarizes the possible outcomes for a true/false followed by a multiple choice question. Note that there are 10 possible combinations. A tree diagram is the illustration of all the possible outcomes of a procedure. These diagrams are helpful if you have problems determining probabilities.

4. Tree Diagram of Example #1: T T Thus we have T,T T,H H,T H,H 2 x 2 = 4 total outcomes. H T H H First toss, 2 possible outcomes Second toss, 2 possible outcomes in each

5. 3. Recall a family that want to have three children. What is the probability that they only have one girl? B B G B B G G B B G G B G G

6. 4. A box contains 5 red markers and 7 black markers. We select 2 markers at random. Find the probability of choosing a red marker first and a black marker second. P(red first)=5/12. What about P(black second)? This situation is just a little bit complicated. Remember that we are selecting markers from a box of 12, but we will need to take into consideration that we have already selected 1 marker and that marker happened to be red. In a situation like this, we will have to either a. Sample with replacement. b. Sample without replacement. a. Sample with replacement b. Sample without replacement

7. 5. In a standard deck of cards, find the probability of selecting a heart first, a spade second and a club third. a. With replacement b. Without replacement One thing that you should notice in doing problems with replacement is that they have one thing in common. The two events are independent of one another. The problems done without replacement are considered dependent. Independent Events: Two events A and B are independent if the occurrence of one event does not affect the probability of the occurrence of the other event. If the events are not independent, they are said to be dependent.

8. Formulas for the Multiplication Rule: We will focus on independent events. Where A is the first event and B is the second event.

9. 6. A professor decides to give a 5 question multiple choice quiz consisting of A, B, C, D. Find the probability of getting the first 3 correct and the last two incorrect. 7. A space ship must be lifted into outer space with the assistance of four rocket boosters. If a rocket booster has a defect rate of 1.5%, what is the probability that the space ship will be lifted into outer space with the aid of four independent boosters?

10. 8. A student in the class was born on October 31. What is the probability that two randomly selected students were born on October 31, ignoring a leap year. 9. What is the probability of randomly selecting five people and they all have the same birth date?

11. Conditional Probability A Conditional Probabilityof an event is a probability obtained with the additional information that some other event has already occurred. P(B |A) denotes the Conditional Probability of event B occurring, given that event A has already occurred. It can be found by the following formula. Our intuitive approach to conditional probability is a follows: The conditional probability of B given A can be found by assuming that event A has occurred and then working under that assumption to calculate the probability that event B will occur.

12. 1. Back to our table Find the probability that someone died of heart disease given that they were a smoker.

13. 2. The table below summarizes results from a survey of 120 randomly selected college students. It states gender and political affiliation. a. Find the probability that someone is a male given that they are a Democrat.

14. 3. Suppose 100 employees of a company were asked whether they are in favor or against paying CEOs of U.S. companies. The table below summarizes the data. Determine

15. Complements & At Least One The Probability of “At Least One” is a procedure of calculating a probability that uses the multiplication rule and complements together. Key Vocabulary: • At Least One = One or More • The complement of At Least One = none Example: Recall the sample space of having 3 children BBBBBGBGBBGGGBBGBGGGBGGG At least one Boy The complement of At least one Boy=No Boys=All Girls

16. To find the probability of At Least One of E: • Calculate the probability of the complement of E • Raise the complement of E to the number of trials • Subtract that result from 1. • P(at least one of A) = 1–[P(Complement of A)]n • where n is the number of trials • 1. Let us find the probability of getting at least one boy among three births, but with out using the table. P( no boys out of 3 births) Which happens to be the same as using the table.

17. 2. You take a 5 question multiple choice quiz (A,B,C,D). What is the probability of getting at least one correct on the quiz. 3. With one method of acceptance sampling, a sample of items is randomly selected and the entire batch is rejected if there is at least one defect. A company that manufactures satellite radios has just manufactured its new line of radios. The defect rate of a radio is 1.5%. A batch of 10 radios are randomly selected and tested, what is the probability that the entire batch will be rejected?