Section 7.2 Definition of Probability
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1 Section 7.2 Definition of Probability Question: Suppose we have an experiment that consists of flipping a fair 2-sided coin and observing if the coin lands on heads or tails? From section 7.1 weshouldknowthatthereare outcomes, and that only the coin landing on heads? Answer: of those outcomes is a head, but what is the probability of Probability of heads = #ofoutcomeswithahead #oftotaloutcomes Probability of an Event E: Part I If S is the sample place of an experiment with event E, then the probability of event E occuring, written as P (E), is given by P (E) = n(e) n(s) 1. A family has three children. Assuming a boy is as likely as a girl to have been born, what are the following probabilities? (a) Three are boys and none are girls. (b) At least 2 are girls.
2 2. A pair of fair 6-sided dice is rolled. What is the probability of each of the following? (Round your answers to three decimal places.) (a) The sum of the numbers shown uppermost is less than 6 (b) At least one 3 is cast 3. In a sweepstakes sponsored by Gemini Paper Products, 10, 000 entries have been received. If 1 grand prize is drawn, and 4 first prizes, 30 second prizes, and 550 third prizes are to be awarded, what is the probability that a person who has submitted one entry will win the following? (a) a third prize (Round your answer to four decimal places if necessary.) (b) any prize (Round your answer to four decimal places if necessary.) 2 Fall 2017, Maya Johnson
3 4. If a card is drawn at random from a standard 52-card deck, what is the probability that the card drawn is one of the following? (a) aclub (b) ablackcard (c) aking 5. In an attempt to study the leading causes of airline crashes, the following data were compiled from records of airline crashes (excluding sabotage and military action): Primary Factor Accidents Flight crew 323 Airplane 47 Maintenance 12 Weather 25 Airport/air tra c control 25 Miscellaneous/other 14 Assume that you have just learned of an airline crash and that the data give a generally good indication of the causes of airplane crashes. Give an estimate of the probability that the primary cause of the crash was due to flight crew or bad weather. (Round your answer to three decimal places.) 3 Fall 2017, Maya Johnson
4 Probability Distribution If S = {s 1,s 2,,s n } is the sample space of a given experiment, then the probability distribution is a table where the entries in the first row are all the outcomes of S and the entries in the second row are all their corresponding probabilities. Outcomes s 1 s 2 s n Probability P (s 1 ) P (s 2 ) P (s n ) 6. The grade distribution for a certain class is shown in the following table. Find the probability distribution associated with these data. (Enter your answers to three decimal places.) Grade A B C D F Frequency of Occurence Probability 7. In a survey conducted by a business-advisory firm of 4980 adults 18 years old and older in June 2009, during the Great Recession, the following question was asked: How long do you think it will take to recover your personal net worth? The results of the survey follow. (Round your answers to three decimal places.) Answer (years) Respondents (a) Determine the empirical probability distribution associated with these data. Answer (years) Probability (b) If a person who participated in the survey is selected at random, what is the probability that he or she expects that it will take 5 or more years to recover his or her personal net worth? 4 Fall 2017, Maya Johnson
5 Properties of Probability Distributions 1. 0 apple P (s i ) apple 1fori =1, 2,,n. 2. P (s 1 )+P(s 2 )+ + P (s n )=1 3. P ({s i }[{s i })=P(s i )+P(s i )(i6= j) fori =1, 2,,n; j =1, 2,,n. Finding the Probability of an Event E: PartIIIf E = {s 1,s 2,,s m } is an event of a sample space S of a given experiment, then P (E) =P (s 1 )+P (s 2 )+ + P (s m ) If E is the empty set,?, thenp (E) =0. 8. Adieisloaded,andithasbeendeterminedthattheprobabilitydistributionassociatedwith the experiment of rolling the die and observing which number falls uppermost is given by the following: Simple Event Probability {1}.18 {2}.13 {3}.19 {4}.2 {5}.15 {6}.15 (a) What is the probability of the number being even? (b) What is the probability of the number being either a 1 or a (c) What is the probability of the number being less than 4? 5 Fall 2017, Maya Johnson
6 9. Let S = {s 1,s 2,s 3,s 4,s 5,s 6 } be the sample space associated with the experiment having the following probability distribution. (Enter your answers as fractions.) Outcomes s 1 s 2 s 3 s 4 s 5 s 6 Probability 1 12 (a) Find the probability of A = {s 1,s 3 } (b) Find the probability of B = {s 2,s 4,s 5,s 6 }. (c) Find the probability of C = S. 10. The sample space S = {s 1,s 2,s 3 } has the property that P (s 1 )+P (s 2 )=0.39 and P (s 2 )+P (s 3 )= Find the probability of each outcome. 6 Fall 2017, Maya Johnson
7 Section 7.3 Rules of Probability Definition: Two events E and F are said to be mutually exclusive if the two events have no outcomes in common, that is E \ F =?. Properties of Probabilities: Revisited 1. 0 apple P (E) apple 1foranyeventE. 2. P (S) =1. 3. If E and F are mutually exclusive, then P (E [ F )=P(E)+P(F). 4. If E and F are any two events of an experiment, then P (E [ F )=P(E) +P (F ) P (E \ F ). (This should remind us of the union/addition rule from section 6.2.) 5. If E is an event of an experiment and E c denotes the complement of E, thenp (E c )=1 P (E). The reverse is also true, namely, P (E) =1 P (E c ). (I call this the Complement Rule) 1. An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event Adiamondorakingisdrawn. 2. Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.59, P (F )=0.38, and P (E c \ F )=0.28. Calculate the probabilities below. (a) P (E c ) (b) P (E \ F ) (c) P (E [ F ) (d) P (E c \ F c ) (e) P (E [ F c ) 7 Fall 2017, Maya Johnson
8 3. Let E and F be two mutually exclusive events, and suppose P (E) = 0.6 andp (F ) = 0.1. Compute the probabilities below. (a) P (E \ F ) (b) P (E [ F ) (c) P (E c ) (d) P (E c \ F c ) (e) P (E c [ F c ) 4. Let S = {s 1,s 2,s 3,s 4,s 5,s 6 } be the sample space associated with an experiment having the following partial probability distribution. Outcomes s 1 s 2 s 3 s 4 s 5 s 6 Probability 4 29 Consider the events: A = {s 1,s 2,s 5 }, B = {s 3,s 5,s 6 }, C = {s 1,s 3,s 4,s 6 },andd = {s 1,s 2,s 3 } Calculate the following probabilities. (Give answers as fractions.) (a) P (s 2 ) (b) P (D) (c) P (B c ) (d) P (A c \ B) (e) P (C c [ D) Fifa t ate=# 8 Fall 2017, Maya Johnson
9 5. Among 500 freshmen pursuing a business degree at a university, 317 are enrolled in an economics course, 214 are enrolled in a mathematics course, and 138 are enrolled in both an economics and a mathematics course. What is the probability that a freshman selected at random from this group is enrolled in each of the following? (Round answers to three decimal places.) (a) an economics or a mathematics course (b) exactly one of these two courses (c) neither an economics course nor a mathematics course 6. The following table gives the percentage of music downloaded from the United States and other countries by U.S. users: Country U.S. Germany Canada Italy U.K. France Japan Other Percent (Round answer to three decimal places.) (a) Verify that the table does give a probability distribution for the experiment. (b) What is the probability that a U.S. user who downloads music, selected at random, obtained it from either the United States or Canada? (c) What is the probability that a U.S. user who downloads music, selected at random, does not obtain it from Italy, the United Kingdom (U.K.), or France? 9 Fall 2017, Maya Johnson
10 7. Apollwasconductedamong250residentsofacertaincityregardingtoughergun-controllaws. The results of the poll are shown in the table. (Round answers to three decimal places.) Own Own Own a Only a Only a Handgun Own Handgun Rifle and a Rifle Neither Total Favor Tougher Laws Oppose Tougher Laws No Opinion Total (a) If one of the participants is selected at random, what is the probability that he or she favors tougher gun-control laws? (b) If one of the participants is selected at random, what is the probability that he or she owns ahandgun? (c) If one of the participants is selected at random, what is the probability that he or she owns ahandgunbutnotarifle? (d) If one of the participants is selected at random, what is the probability that he or she favors tougher gun-control laws and does not own a handgun? 10 Fall 2017, Maya Johnson
11 Section 7.4 Use of Counting Techniques in Probability Question: Five marbles are selected at random without replacement from a jar containing four white marbles and six blue marbles. From Section 6.4, we know that there are ways to choose these five marbles. We should also know that of those samples have all blue marbles. How do we find the probability that our sample has all blue marbles? Computing the Probability of an Event in a Uniform Sample Space: Revisited Let S be a uniform sample space, and let E be any event. Then P (E) = Number of outcomes in E Number of outcomes in S = n(e) n(s) Let s revisit the above question: 1. Five marbles are selected at random without replacement from a jar containing four white marbles and six blue marbles. Find the probability of the given event. (Round answer to three decimal places.) All of the marbles are blue. 2. A 4-card hand is drawn from a standard deck of 52 playing cards. Find the probability that the hand contains the given cards. (Round answer to 3 decimal places.) No diamonds. =.3@ 3. Three cards are selected at random without replacement from a well-shu ed deck of 52 playing cards. Find the probability of the given event. (Round answer to four decimal places.) Three cards of the same suit are drawn. I# 11 Fall 2017, Maya Johnson
12 4. Aboxhas7marbles,3ofwhicharewhiteand4ofwhicharered. Asampleof4marblesis selected randomly from the box without replacement. (Give answers as an exact fraction.) (a) What is the probability that exactly 2 are white and 2 are red? (b) What is the probability that at least 2 of the marbles are white? 5. Jacobs & Johnson, an accounting firm, employs 20 accountants, of whom 6 are CPAs. If a delegation of 4 accountants is randomly selected from the firm to attend a conference, what is the probability that 4 CPAs will be selected? (Round answer to three decimal places.) = :D 6. AshelfintheMetroDepartmentStorecontains90coloredinkcartridgesforapopularink-jet printer. Eight of the cartridges are defective. (Round answers to five decimal places.) (a) If a customer selects 2 cartridges at random from the shelf, what :@ is the probability that both are defective? 12 Fall 2017, Maya Johnson
13 (b) If a customer selects 2 cartridges at random from the shelf, what is the probability that at least 1 is defective? =.17@ 7. AcustomerfromCavallaro sfruitstandpicksasampleof4orangesatrandomfromacrate containing 65 oranges, of which 5 are rotten. What is the probability that the sample contains 1 or more rotten oranges? (Round answer to three decimal places.) : 8. Astudentstudyingforavocabularytestknowsthemeaningsof14wordsfromalistof26words. If the test contains 10 words from the study list, what is the probability that at least 8 of the words on the test are words that the student knows? (Round answer to three decimal 13 Fall 2017, Maya Johnson
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