HW 5.1-5.5

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If E and F are disjoint​ events, then P(E or F)=

P(E)+P(F)

If E and F are not disjoint​ events, then​ P(E or F)=​________.

P(E)+P(F)-P(E and F)

How many different​ 10-letter words​ (real or​ imaginary) can be formed from the following​ letters? WRELPOCTKV

10!/2! = 10x9x8x7x6x5x4x3 = 1814400

A(n) ____________ is an ordered arrangement of r objects chosen from n distinct objects without repetition.

permutation

A baseball player hit 65 home runs in a season. Of the 65 home​ runs, 21 went to right​ field, 21 went to right center​ field, 11 went to center​field, 10 went to left center​ field, and 2 went to left field. ​(a) What is the probability that a randomly selected home run was hit to right​ field? ​(b) What is the probability that a randomly selected home run was hit to left​ field? ​(c) Was it unusual for this player to hit a home run to left​ field? Explain.

(a) 21/65=0.323 (b) 0.031 (c) Yes, because left field is less than <0.05

A cheese can be classified as either raw-milk or pasteurized. Suppose that 90​% of cheeses are classified as pasteurized. ​(a) Two cheeses are chosen at random. What is the probability that both cheeses are pasteurized​? ​(b) Five cheeses are chosen at random. What is the probability that all five cheeses are pasteurized​? ​(c) What is the probability that at least one of five randomly selected cheeses is raw-milk​? Would it be unusual that at least one of five randomly selected cheeses is raw-milk​?

(a) 0.90(0.90)=0.81 (b) 0.90^5=0.5905 (c) 1-0.5905=0.4095 It would also not be unusual.

The probability that a randomly selected 2​-year-old male salamander will live to be 3 years old is 0.96523. ​(a) What is the probability that two randomly selected 2​-year-old male salamanders will live to be 3 years​ old? ​(b) What is the probability that seven randomly selected 2​-year-old male salamanders will live to be 3 years​ old? ​(c) What is the probability that at least one of seven randomly selected 2​-year-old male salamanders will not live to be 3 years​ old? Would it be unusual if at least one of seven randomly selected 2​-year-old male salamanders did not live to be 3 years​ old?

(a) 0.93167 (b) 0.78058 (c) 0.21942 (d)No because the probability of this happening is greater that 0.05

The probability that a randomly selected 5​-year-old male chipmunk will live to be 6 years old is 0.95773. ​(a) What is the probability that two randomly selected 5​-year-old male chipmunks will live to be 6 years​ old? ​(b) What is the probability that seven randomly selected 5​-year-old male chipmunks will live to be 6 years​ old? ​(c) What is the probability that at least one of seven randomly selected 5​-year-old male chipmunks will not live to be 6 years​ old? Would it be unusual if at least one of seven randomly selected 5​-year-old male chipmunks did not live to be 6 years​ old?

(a) 0.95773*=0.91725 (b) 0.95773^7=0.73910 (c) 1-0.73910=0.2609 (d) No ..........Greater than 0.05

A test to determine whether a certain antibody is present is 99.8​% effective. This means that the test will accurately come back negative if the antibody is not present​ (in the test​ subject) 99.8​% of the time. The probability of a test coming back positive when the antibody is not present​ (a false​ positive) is 0.002. Suppose the test is given to four randomly selected people who do not have the antibody. ​(a) What is the probability that the test comes back negative for all four ​people? ​(b) What is the probability that the test comes back positive for at least one of the four ​people?

(a) 0.992 (b) 0.008

A bag of 35 tulip bulbs contains 14 red tulip​ bulbs, 12 yellow tulip​ bulbs, and 9 purple tulip bulbs. Suppose two tulip bulbs are randomly selected without replacement from the bag. ​(a) What is the probability that the two randomly selected tulip bulbs are both​ red? ​(b) What is the probability that the first bulb selected is red and the second​ yellow? ​(c) What is the probability that the first bulb selected is yellow and the second​ red? ​(d) What is the probability that one bulb is red and the other​ yellow?

(a) 14/35 x 13/34 = 0.153 (b) 14/35 x 12/34 = 0.141 (c) 12/35 x 14/34 = 0.141 (d) 0.141 + 0.141 = 0.282

Suppose you just purchased a digital music player and have put 9 tracks on it. After listening to them you decide that you like 2 of the songs. With the random feature on your​ player, each of the 9 songs is played once in random order. Find the probability that among the first two songs played ​(a) You like both of them. Would this be​ unusual? ​(b) You like neither of them. ​(c) You like exactly one of them. ​ Redo​ (a)-(c) if a song can be replayed before all 7 songs are played (d) The probability that you like both songs is The probability that you like neither song is The probability that you like exactly one song is

(a) 2/9 x 1/8 =0.028 (yes unusual) (b)7/9 x 6/8 = 0.583 (c)2/9 x 7/8 + 7/9 x 2/8 = 0.389 (d) 2/9 x 2/9 = 0.049 7/9 x 7/9 = 0.605 2/9 x 7/9 + 7/9 x 2/9 = 0.346

In a recent​ poll, a random sample of adults in some country​ (18 years and​ older) was​ asked, "When you see an ad emphasizing that a product is​ "Made in our​ country," are you more likely to buy​ it, less likely to buy​ it, or neither more nor less likely to buy​ it?" The results of the​ survey, by age​ group, are presented in the following contingency table. Complete parts​ (a) through​ (c). (a) What is the probability that a randomly selected individual is 35 to 44 years of​ age, given the individual is more likely to buy a product emphasized as​ "Made in our​ country"? (b) What is the probability that a randomly selected individual is more likely to buy a product emphasized as​ "Made in our​ country," given the individual is 35 to 44 years of​ age? ​(c) Are​ 18- to​ 34-year-olds more likely to buy a product emphasized as​ "Made in our​ country" than individuals in​ general?

(a) 309/1328=0.233 (b) 309/563=0.59 (c) No, Less likely

A certain four​-cylinder combination lock has 45 numbers on it. To open​ it, you turn to a number on the first​ cylinder, then to a second number on the second​ cylinder, and then to a third number on the third cylinder and so on until a four​-number lock combination has been effected. Repetitions are​ allowed, and any of the 45 numbers can be used at each step to form the combination.​ (a) How many different lock combinations are​ there? (b) What is the probability of guessing a lock combination on the first​ try?

(a) 45^4 4100625 (b) 1/4100625

The grade appeal process at a university requires that a jury be structured by selecting seven individuals randomly from a pool of eight students and seven faculty.​ (a) What is the probability of selecting a jury of all​ students? (b) What is the probability of selecting a jury of all​ faculty? (c) What is the probability of selecting a jury of four students and three faculty?

(a) 8C7=8/6435 =0.00124 (b) 7C7=1/6435 =0.00016 (c) 0.38073

Determine whether the events E and F are independent or dependent. Justify your answer. ​(a) ​E: A person attaining a position as a professor. ​F: The same person attaining a PhD. (b) ​E: A randomly selected person accidentally killing a spider. ​F: A different randomly selected person accidentally swallowing a spider. (c) ​E: The rapid spread of a cocoa plant disease. ​F: The price of chocolate.

(a) E and F are dependent because attaining a PhD can affect the probability of a person attaining a position as a professor. (b) E cannot affect F and vice versa because the people were randomly​ selected, so the events are independent. (c) The rapid spread of a cocoa plant disease could affect the price of chocolate​, so E and F are dependent.

For a parallel structure of identical​ components, the system can succeed if at least one of the components succeeds. Assume that components fail independently of each other and that each component has a 0.16 probability of failure. Complete parts​ (a) through​ (c) below. ​(a) Would it be unusual to observe one component​ fail? Two​ components? (b) What is the probability that a parallel structure with 2 identical components will​ succeed? (c) How many components would be needed in the structure so that the probability the system will succeed is greater than 0.9998​?

(a) It would not be unusual to observe one component​ fail, since the probability that one component​ fails, 0.16​, is greater than 0.05. It would be unusual to observe two components​ fail, since the probability that two components​ fail, 0.0256 is less than 0.05. (b) 0.9744 (c) 5

The following data represent the number of different communication activities used by a random sample of teenagers in a given week. Complete parts​ (a) through​ (d). (a) Are the events "male​" and "5+ activities" independent? (b) ​(b) Are the events "female​" and "0 activities" independent? ​(c) Are the events "1−2 activities" and "5+activities" mutually​ exclusive? (d) Are the events "male​" and "1−2 activities" mutually​ exclusive?

(a) No, because Upper P left parenthesis male right parenthesis and Upper P left parenthesis male| 5 plus activities right parenthesisP(male) and P(male|5+ activities) are not equal. (b) Yes, because Upper P left parenthesis female right parenthesis and Upper P left parenthesis female| 0 activities right parenthesisP(female) and P(female|0 activities) are equal. (c) Yes, because Upper P left parenthesis 1 minus 2 activities and 5 plus activities right parenthesisP(1−2 activities and 5+ activities is zero. (d) No, because Upper P left parenthesis male and 1 minus 2 activities right parenthesisP(male and 1−2 activities) is not zero.

According to a​ poll, about 16​% of adults in a country bet on professional sports. Data indicates that 46.6​% of the adult population in this country is male. Complete parts​ (a) through​ (e). (a) (a) Are the events​ "male" and​ "bet on professional​ sports" mutually​ exclusive? Explain. (b) Assuming that betting is independent of​ gender, compute the probability that an adult from this country selected at random is a male and bets on professional sports. (c) Using the result in part​ (b), compute the probability that an adult from this country selected at random is male or bets on professional sports. (d) The poll data indicated that 9.6​% of adults in this country are males and bet on professional sports. What does this indicate about the assumption in part​ (b)? (e) How will the information in part​ (d) affect the probability you computed in part​ (c)? Select the correct choice below and fill in any answer boxes within your choice.

(a) No. A person can be both male and bet on professional sports at the same time. (b) .466x.16=0.0746 (c) .466+.16=0.626-0.0746 =0.5514 (d) The assumption was incorrect and the events are not independent. (e) 0.466+0.16-0.096=0.5300

What is the probability of an event that is​ impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is​ impossible? (a) What is the probability of an event that is​ impossible? (b) Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is​ impossible?

(a) ZERO (0) (b) NO

List all the permutations of three objects a, b, and c taken two at a time without repetition. What is 3P2​?

(a) ab, ac, ba, bc, ca, cb (b) 6

List all the combinations of three objects x, y, and z taken two at a time. What is 3C2​?

(a) xy, xz, yz, (b) 3

According to a certain​ country's department of​ education, 40.5​% of​ 3-year-olds are enrolled in day care. What is the probability that a randomly selected​ 3-year-old is enrolled in day​ care?

0.405

Suppose that E and F are two events and that P(E and F)=0.2 and P(E)=0.4. What is P(F|E)​?

0.5

If P(E)=0.45​, P(E or F)=0.65​, and​ P(E and F)=0.10​, find​ P(F).

0.65-0.45=0.20 0.20+0.10=0.30

9C8=

9!/8! (9-8)! = 9!/8!(1)!=9

Suppose Jim is going to build a playlist that contains 14 songs. In how many ways can Jim arrange the 14 songs on the​ playlist?

14! = 87178291200

Four members from a 27​-person committee are to be selected randomly to serve as​ chairperson, vice-chairperson,​ secretary, and treasurer. The first person selected is the​ chairperson; the​ second, the​ vice-chairperson; the​ third, the​ secretary; and the​ fourth, the treasurer. How many different leadership structures are​ possible?

27!/(27-4)!=27!/23!=421200

Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E={1, 3, 5, 6}.

4/10=2/5=.40

8P2 =

8!/(8-2)! = 8!/6! =8X7=56

A golf ball is selected at random from a golf bag. If the golf bag contains 6 green ​balls, 2 brown ​balls, and 13 black ​balls, find the probability of the following event. The golf ball is green or brown.

8/21=0.381

A woman has nine skirts and four blouses. Assuming that they all​ match, how many different skirt​-and-blouse combinations can she ​wear?

9 x 4 =36

The notation P(F E) means the probability of event ________ given event _______.

F given event E

Two events E and F are​ ________ if the occurrence of event E in a probability experiment does not affect the probability of event F.

Independent

You suspect a​ 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the following table. Do you think the die is​loaded? Why?

No because each value has an approximately equal chance of occurring.

A probability experiment is conducted in which the sample space of the experiment is S={7,8,9,10,11,12,13,14,15,16,17,18}. Let event E={8,9,10,11,12,13} and event F={12,13,14,15}. List the outcomes in E and F. Are E and F mutually​exclusive?

{12,13} No E and F has outcomes in common

In a certain card​ game, the probability that a player is dealt a particular hand is 0.43. Explain what this probability means. If you play this card game 100​ times, will you be dealt this hand exactly 43 times? Why or why​ not?

The probability 0.43 means that approximately 43 out of every 100 dealt hands will be that particular hand.​ No, you will not be dealt this hand exactly 43 times since the probability refers to what is expected in the​ long-term, not​short-term.

Why is the following not a probability​ model? Color. Probability Red. 0.2 Green. −0.3 Blue. 0.1 Brown 0.3 Yellow. 0.3 Orange. 0.4

This is not a probability model because at least one probability is less than 0.

Use the given​ table, which lists six possible assignments of probabilities for tossing a coin​ twice, to determine which of the assignments of probabilities are consistent with the definition of a probability model.

​A,B,C,F is/are consistent with the definition of a probability model. A-1/4, 1/4,1/4,1/4 B-0,0,0,1 C-1/9,1/9,7/18,7/19 F-1/8,1/4,1/4,3/8


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