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1 4-1 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting, and in various areas. 4-2 Defns any activity that leads to well-defined results called,the result of a single trial of a probability., -the of all of a probability experiment. a of the sample space. an event with outcome. consists of outcomes. (a) A die is tossed one time. List the elements of the sample space S. (b) List the elements of the event consisting of a number that is greater than 4. Example 2 A coin is tossed twice. List the elements of the sample space S, and list the elements of the event consisting of at least one head. [Using a tree diagram:] Example 3 A randomly selected citizen is interviewed and the following information is recorded: employment status and level of education. The symbols for employment status are Y = employed and N = unemployed, and the symbols for level of education are 1 = did not complete high school, 2 = completed high school but did not complete college, and 3 = completed college.

2 II. Calculating Probabilities 1. -( Approximation of Probability) An experiment is performed, and then the number of times that event A actually. Based on these actual results, P(A) is estimated by the of A. P(A) Rounding Rule- Probabilities should be expressed as fractions or rounded to or decimal places. When the probability of an event is an extremely decimal, it is permissible to round the decimal to the first digit after the decimal point. Example 1 : The age distribution of employees for this college is shown below: Age # of employees Under 20 If an employee is selected at, find the probability that he or she is in the following age groups. (a) Between 30 and 39 years of age (b) Under 20 or over 49 years or age Example 2-During a sale at men s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters, and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought a sweater that was. 2. Approach to Probability (Requires Outcomes) Suppose that A is an event in a sample space S and each outcome in S is. Let n(a) represent the number of outcomes in A and n(s) represent the total number of outcomes in S. Then P(A)=

3 Example 1 A statistics class contains 14 males and 20 females. A student is to be selected by chance and the gender of the student recorded. (a) Use the Classical Approach to Probability to find P(M) and P(F). S= P(M)= P(F)= (b) Is each outcome equally likely? Explain. (C) Use the Empirical Approach to find P(M) and P(F) based on the data obtained. Example: Two dice are tossed. Find the probability that the sum of two dice is greater than 8? S= Example 3 : If one card is drawn from a deck, find the probability of getting (a) a club (b) a 4 and a club

4 Example 4 Three equally qualified runners, Mark (M), Bill (B) and Alan (A), run a 100-meter, sprint, and the order of finish is recorded. (a) Give a sample space S. (b) What is the probability that Mark will finish last? uses a probability value based on an educated or, employing opinions and information. In probability, a person or group makes an educated guess at the that an event will occur. This guess is based on the person s experience and evaluation of a solution. III. Requirements for Probabilities 1.) 2.) Probabilities should be expressed as fractions or rounded to or decimal places. Example 1: A probability experiment is conducted. Which of these can be considered a probability of an outcome and why? (a) 2 / 5 (b) (c) 1.09 Example 2: Given: S = {E1, E2, E3, E4}, and P(E1) = P(E2) = 0.2 and P(E3) = 0.5, find P(E4)

5 IV. The Complement Rule The of event A is the set of in the sample space that are included in the outcomes of event A. For any event A, the probability of its complement is Example 1 The chance of raining tomorrow is 70%. What is probability that it will not rain tomorrow? 4-2 The Addition Rules for Probability I. Addition Rule for Mutually Exclusive Events Event A and B are if they occur together when the experiment is performed. (e.g. no elements) If A, B are mutually exclusive events, then P (A or B) = Example 1 A single card is drawn from a deck. Find the probability of selecting a club or a diamond. Example 2 In a large department store, there are 2 managers, 4 department heads, 16 clerks, and 4 stock persons. If a person is selected at random, find the probability that the person is either a clerk or a manager. II. General Addition Rule If A, and B are mutually exclusive events, then P (A or B) = Example 1 A single card is drawn from a deck. Find the probability of selecting a jack or a black card.

6 Example 2 In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comics section and other do not. The distribution is shown here. Yes No Having comics Section Morning Evening Weekly If a newspaper is selected at random, find these probabilities. (a) The newspaper is weekly publication. (b) The newspaper is a daily morning publication or has comics. (c) The newspaper is published weekly or does not have comics.

7 Section 4 3 I. Multiplication Rule for Independent Events Events A and B are if the outcome of event A has on the outcome of event B. Suppose A and B are independent events, then P (A and B) = Example 1: If 36% of college students are overweight, find the probability that if three college students are selected at random, all will be overweight. Example 2: If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will be U.S. citizens. Example 3:Suppose the probability of remaining with a particular company 10 years or longer is 1/6. A man and a woman start work at the company on the same day. (a) What is the probability that the man will work there less than 10 years? (b) What is the probability that both the man and woman will work there less than 10 years? Example 4: A smoke-detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is 0.95; by device B, 0.98; and by both devices, (a) If smoke is present, find the probability that the smoke will be detected by device A or device B or by both devices. (b) Find the probability that the smoke will not be detected.

8 Example 5:If you make random guesses for four multiple-choice test questions (each with five possible answers), what is the probability of getting at least one correct? Example 6: There are 2000 voters in a town. Consider the experiment of randomly selecting a voter to be interviewed. The event A consists of being in favor of more stringent building codes; the event B consists of having lived in the town less than 10 years. The following table gives the number of voters in various categories. a.) b.) c.) II. Multiplication Rule for Dependent Events Events A and B are if the outcome of event A the outcome of event B. Suppose A and B are dependent events, then P (A and B) = Example 1:A box has 5 red balls and 2 white balls. If two balls are randomly selected (one after the other), what is the probability that they both are red? (a) With replacement (independent case) b) Without replacement (dependent case)

9 Ex 2: Three cards are drawn from a deck without replacement. Find the probability that all are Jacks. III. Conditional Probability The probability that B occurs that A has can be found by the probability that events occurred by the probability that the first event (A) has occurred. P(B A) = Example 1: Two fair dice are tossed. Consider the following events. A = sum is 7 or more, B = sum is even, C = a match (both numbers are the same). S = the sample space is the same as we did previously Find the events in the sets A, B, and C. What is P(A), P(B), P(C)? a.) What is P(A and B)? b.) What is P (A or B) (c) P (A B) = (d) P (B C) =

10 Example 2 :At a local Country Club, 65% of the members play bridge and swim, and 72% play bridge. If a member is selected at random, find the probability that the member swims, given that the member plays bridge. Example 3 : Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table. Class Favor Oppose No opinion Freshman Sophomore If a student is selected at random, find these probabilities. (a) The student is a freshman or favors the ban. (b) Given that the student favors the ban, the student is a sophomore. Section 4-4 I. Principle of Counting If Task 1 can be done in n ways and Task 2 in m ways, Task 1 and Task 2 performed together can be done in. (Also called the Multiplication Rule) Example 1 : Two dice are tossed. How many outcomes are in S. Example 2 :A password consists of two letters followed by one digit. How many different passwords can be created? (Note: Repetitions are allowed) Example 3 : Suppose four digits are to be randomly selected (with repetitions allowed). (Note: the set of digits is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.) If the first digit must be a 2 and repetitions are permitted, how many different 4-digit can be made?

11 Permutations The number of ways we can n objects, them r at a time, is : npr = Combinations The number of combinations of n objects can be formed, them r at a time is ncr= Note: Combinations are always than permutations for the same n and r. Example 1 : a.) b.) c.) d.) Example 2: An inspector must select 3 tests to performed in a certain order on a manufactured part. He has a choice of 7 tests. How many ways can be performed 3 different tests? Example 3: If a person can select 3 presents from 10 presents, how many different combinations are there? Example 4: Your family vacation involves a cross-country air flight, a rental car, and a hotel stay in Boston. If you can choose from four major air carriers, five car rental agencies, and three major hotel chains, how many options are available for your vacation accommodations?

12 Section 4 5 I. Probability and Counting Rules Use the multiplication rule, combination and permutation learned from Section 4-5 to the number of ways to a given event, n(a), and to the number of ways for the of an experiment, n(s). The probability of A = P(A) = Ex 1: A combination lock consists of 26 letters of the alphabet. If a three-letter combination is needed, find the probability that the combination will consist of the first two letters AB in that order. The same letter can be used more than once. Ex 2: Five cards are selected from a 52-card deck for a poker hand. (a) How many outcomes are in the sample space? b.) A royal flush is a hand that contains that A, K, Q, J, 10, all in the same suit. How many ways are there to get a royal flush? c) What is the probability of being dealt a royal flush?

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