1. + Chapter 5 Probability: What Are the Chances? 5.1Randomness, Probability, and Simulation 5.2Probability Rules 5.3Conditional Probability and Independence Section 5.2 Probability Rules After this section, you should be able to… DESCRIBE chance behavior with a probability model DEFINE and APPLY basic rules of probability DETERMINE probabilities from two-way tables CONSTRUCT Venn diagrams and DETERMINE probabilities Learning Objectives

2. Sample Space - the collection of all possible outcomes of a chance experiment Roll a dieS = {1,2,3,4,5,6} Event - any collection of outcomes from the sample space Complement - Consists of all outcomes that are _____in the event - Not rolling a even # = 1- P(rolling an even #)

3. Union - the event A ____ B happening consists of all outcomes that are in at least one of the two events - Rolling a prime # or even numberE={2,3,4,5,6} Intersection - the event A _____ B happening consists of all outcomes that are in both events - Drawing a red card and a “2” E = {2 hearts, 2 diamonds}

4. Addition rule: If A and B are disjoint events, then _____________________ If A and B are NOT disjoint events, then ________________________________ Mutually Exclusive (disjoint) -two events that have ______________________ - Roll a “2” or a “5”

5. At least one: The probability that at least one outcome happens is ___________________________ P(at least 1) = 1 – P(none) Multiplication rule: If there are n ways to do a first event & m ways to do a second event. Then the number of all possible outcomes = _____

6. Disjoint example: A large auto center sells cars made by many different manufacturers. Three of these are Honda, Nissan, and Toyota. Suppose that P(H) =.25, P(N) =.18, P(T) =.14. Are these disjoint events? P(H or N or T) = P(not (H or N or T) =

7. NonDisjoint example: Musical styles other than rock and pop are becoming more popular. A survey of college students finds that the probability they like country music is.40. The probability that they liked jazz is.30 and that they liked both is.10. What is the probability that they like country or jazz? Are these disjoint events?

8. For a sales promotion the manufacturer places winning symbols under the caps of 10% of all Dr. Pepper bottles. You buy a six-pack. What is the probability that you win something? P(at least one winning symbol) = 1 – P(no winning symbols)

9. Rules of probability SUMMARY What is the range of values for any given probability? What is the sum of the probability of all events? Compliment rule: For any event A, P(A c ) = Addition rule: If A and B are disjoint events, then If A and B are NOT disjoint events, then P(A or B ) =  Multiplication rule: If there are n ways to do a first event & m ways to do a second event. Then the number of all possible outcomes = At least one: P(at least 1) = P(A or B ) =

10. Simple Probability – Example # 1 License plates in the State of Altered require 2 letters followed by 4 digits. No two letters can be the same, but it is ok to repeat digits. How many different license plates can me made?

11. + Simple Probability example #2 : Roll the Dice You are rolling two fair, six-sided dice – one that’s red and one that’s green.How many possible outcomes are there? Probability Rules

12. + Suppose event A is defined as the “sum of 5”. Find P(A). Suppose event B is defined as the “sum is not 5”. Find P(B).

13. + Simple Probability Example #3: Distance Learning Distance-learning courses are rapidly gaining popularity among college students. Randomly select an undergraduate student who is taking distance-learningcourses for credit and record the student’s age. Here is the probability model: Age group (yr):18 to 2324 to 2930 to 3940 or over Probability:0.570.170.140.12 (a)Show that this is a legitimate probability model. (b)Find the probability that the chosen student is not in the traditional college age group (18 to 23 years).

14. Simple Probability Example #4 Canada has 2 official languages, English and French. Choosing a Canadian at random in a recent survey gave the following distribution of responses: EnglishFrenchAsian/PacificOther 0.630.220.06? What is the probability that a Canadian’s mother tongue is either French or Asian Pacific? What is the probability of ‘Other’? Why? What is the probability that a Canadian’s “mother tongue” is not English?

15. + Example #5: Two-Way Tables When finding probabilities involving two events, a two-way table can display the sample space in a way that makes probability calculations easier. Consider the example on page 303. Suppose we choose a student at random. Find the probability that the student Probability Rules (a)has pierced ears. (b)is a male with pierced ears. (c)is a male or has pierced ears. Define events A: is male and B: has pierced ears.

16. Tree Diagrams - When there are many options, you may want to use a TREE DIAGRAM to help you find all the different combinations of outcomes. Simple Probability Example #6 – Four different people are each flipping a coin (4 distinct coin flips). What is the probability of getting... a)At least 2 heads? b) Either 3 heads or 3 tails?

17. Example #6 – Four different people are each flipping a coin (4 distinct coin flips). What is the probability of getting... a) At least 2 heads? b) Either exactly 3 heads or 3 tails?

18. Simple Probability Example #7 – In a family of 3 children… a)What is the probability exactly two are boys? b)What is the probability at least one is a girl?

19. NOT Disjoint: P(A ∩ B) Disjoint: P(A ∪ B) Venn Diagrams

20. + There are 174 total people. You have 90 total males. 19 of them have their ears pierced. You have 103 total people with their ears pierced. Define event A as males Define event B as pierced ears.

21. In an apartment complex, 40% of residents read the USA Today, while 25% of residents read the New York Times. Five percent of residents read both. Suppose we select an apartment resident at random and record which of the two papers the person reads. Venn Diagrams Example #8 Find the probability the person reads at least 1 of the 2 papers. Find the probability the person doesn’t read either paper.

22. After observing several pizza orders, I discover the following pizza topping requests: 67 people wanted sausage on their pizza, 55 wanted pepperoni, 50 wanted mushrooms, 22 wanted sausage and mushrooms, 18 wanted pepperoni and mushrooms, 15 wanted sausage and pepperoni, and 10 wanted all three. What is the probability that a randomly selected person wants a)only sausage? b)pepperoni and mushrooms, but no sausage? c)pepperoni or sausage? d) Neither pepperoni nor sausage nor mushrooms? Venn Diagrams Example #9

23. only sausage? pepperoni and mushrooms, but no sausage? pepperoni or sausage? Sausage Pepperoni Mushrooms d) Neither pepperoni nor sausage nor mushrooms?

24. + Section 5.2 Probability Rules In this section, we learned that… A probability model describes chance behavior by listing the possible outcomes in the sample space S and giving the probability that each outcome occurs. An event is a subset of the possible outcomes in a chance process. For any event A, 0 ≤ P(A) ≤ 1 P(S) = 1, where S = the sample space If all outcomes in S are equally likely, P(A C ) = 1 – P(A), where A C is the complement of event A; that is, the event that A does not happen. Summary

25. + Section 5.2 Probability Rules In this section, we learned that… Events A and B are mutually exclusive (disjoint) if they have no outcomes in common. If A and B are disjoint, P(A or B) = P(A) + P(B). A two-way table or a Venn diagram can be used to display the sample space for a chance process. The intersection (A ∩ B) of events A and B consists of outcomes in both A and B. The union (A ∪ B) of events A and B consists of all outcomes in event A, event B, or both. The general addition rule can be used to find P(A or B): P(A or B) = P(A) + P(B) – P(A and B) Summary

26. + Homework: # 39, 40, 45, 46, 49, 50, 51–54, 56 We’ll learn how to calculate conditional probabilities as well as probabilities of independent events. We’ll learn about Conditional Probability Independence Tree diagrams and the general multiplication rule Special multiplication rule for independent events Calculating conditional probabilities In the next Section…