Stats 515 Online HW 2 and 3

A survey of Amazon.com shoppers reveals the following probability distribution of the number of books per hit: X = 0,1,2,3,4,5,6,7 P(X)=.326,.208,.077,.22,.029,.02,.009,.111 Find the following probabilities: A. What is the probability that an Amazon.com visitor will buy four books? Probability = B. What is the probability that an Amazon.com visitor will buy eight books? Probability = C. What is the probability that an Amazon.com visitor will not buy any books? Probability = D. What is the probability that an Amazon.com visitor will buy at least one book? Probability =

.029 0 .326 .674

Suppose that A and B are two events for which P(A) = 0.23, P(B) = 0.82, and P(A and B) = 0.4 Find P(A|B). P(A|B) =

.4/.82

A fair coin is tossed three times and the events A, B, and C are defined as follows: A : { At least one head is observed } B : { At least two heads are observed } C : { The number of heads observed is odd } Find the following probabilities by summing the probabilities of the appropriate sample points (note that 0 is an even number): (a) P(not B) = (b) P(AandC)= (c) P((notA)orBorC)=

.5 .5 1

According to the U.S. National Center for Edu- cation Statistics, there are more than 63 million American work- ers 18 years old and over who use computers at work. From this study, which was conducted in 1994 and 1998 (Source: Statisti- cal Abstract of the United States, 2000, Table 690), the follow- ing table of joint probabilities was developed. See screenshot of table A. What proportion of workers use a spreadsheet? B. What proportion of male workers use a spreadsheet? C. What proportion of spreadsheet users are female?

.51 .2/.46 .31/.51

x=8,9,10,11,12 P(x)=.1,.2,.1,.1,.5 Given the discrete probability distribution above, determine the following: (a) P(x≥9)= (b) P(x<12)= (c) P(8≤x<10)=

.9 .5 .3

Consider the probablility model with sample space A,B,C and P(A)=0.5, P(B)=0.1, P(C)=0.4.Then (a) P( A or C ) = (b) P( A and B ) =

.9 0

Consider the experiment, called the birthday problem , where our task is to determine the probability that in a group of people ofa certain size there are a least two people who have the same birth- day (the same month and day of month). Suppose there is a room with 51 people in it, find the probability that at least two people have the same birthday. Answer =

.974432

If P(E ∩F) = 0.07, P(E|F) = 0.28, and P(F|E) = 0.7, then (a) P(E)= (b) P(F)= (c) P(E∪F)= (d) Are the events E and F independent? Enter yes or no .

0.1 0.25 0.28 NO

A fair coin is tossed 12 times. What is the prob- ability that: a) Exactly 4 heads appear? b) At least two heads appear? c) At most 9 heads appear?

0.120849609375 0.996826171875 0.980712890625

In the rolling of two fair dice calculate the following: P(Sum of the two dice is 7)= . P(Sum of the two dice is 10)= P(Sum of the two dice is not 9)= P(Sum of the two dice is 4 or 8)= P(Sum of the two dice is not 6 and not 5)=

0.166666666666667 0.0833333333333333 0.888888888888889 0.222222222222222 0.75

Events A1, A2 and A3 form a partiton of the sample space S with probabilities P(A1) = 0.3, P(A2) = 0.2, P(A3) = 0.5. If E is an event in S with P(E|A1) = 0.2, P(E|A2) = 0.3, P(E|A3) = 0.3, compute P(E)= P(A1|E) = P(A2|E) = P(A3|E) =

0.27 0.222222222222222 0.222222222222222 0.555555555555555

Urn A has 16 white and 13 red balls. Urn B has 5 white and 8 red balls. We flip a fair coin. If the outcome is heads, then a ball from urn A is selected, whereas if the outcome is tails, then a ball from urn B is selected. Suppose that a red ball is selected. What is the probability that the coin landed heads?

0.4214463840399

For two events A and B, P(A) = 0.2 and P(B) = 0.4. (a) If A and B are independent, then P(A∪B) = P(A|B) = P(A∩B) = (b) If A and B are dependent and P(A|B) = 0.3, then P(B|A) = P(A∩B) =

0.52 0.2 0.08 0.6 0.12

In a survey of 182 people, the following data were obtained relating gender to political orientation: See table screenshot Republican (R) Democrat (D) Independent (I) Total Male (M) 58 39 15 112 Female (F) 23 28 19 70 Total 81 67 34 182 A person is randomly selected. What is the probability that the person is: a) Male? b) Male and a Democrat? c) Male given that the person is a Democrat? d) Republican given that the person is Male? e) Female given that the person is an Independent? f) Are the events Male and Republican independent? Enter yes or no.

0.615384615384615 0.214285714285714 0.582089552238806 0.517857142857143 0.558823529411765 NO

See tree diagram screenshot Find each probability by referring to the tree diagram above. (a) P(C|A)= (b) P(D|B)= (c) P(A∩C)= (d) P(B∩D)= (e) P(C)= (f) P(D)=

0.8 0.6 0.48 0.24 0.64 0.36

In an experiment, a fair coin is tossed 10 times and the face that appears is recorded. How many elements of the sample space will have no tails ? How many elements of the sample space will have exactly one tail ? How many elements of the sample space will start and end with different faces and have a total of exactly two tails ?

1 10 16

Consider the experiment where a pair of fair dice is thrown. Let X denote the random variable whose value is determined by multiplying the number of spots showing on the one die by the number of spots showing on the other. The range of values that X can assume are the positive integers 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,30,36. Please give the corresponding probabilities for the values of X given below. Pr(X = 1) = Pr(X=2)= Pr(X = 3) = Pr(X = 4) = Pr(X = 5) = Pr(X = 6) = Pr(X = 8) = Pr(X = 9) = Pr(X = 10) = Pr(X = 12) = Pr(X = 15) = Pr(X=16)= Pr(X = 18) = Pr(X = 20) = Pr(X = 24) = Pr(X = 25) = Pr(X = 30) = Pr(X = 36) = Further, find the probability that X is divisible by 12. Probability that X is divisible by 12 equals?

1/36 2/36 2/36 3/36 2/36 4/36 2/36 1/36 2/36 4/36 2/36 1/36 2/36 2/36 2/36 1/36 2/36 1/36 7/36

Consider the experiment where a pair of fair dice is thrown. Let X denote the random variable whose value is determined by taking the minimum of the spots show- ing on either of the two dice thrown. For example, if a 3 and a 5 were thrown, then X would take the value of minimum(3,5) = 3. The range of values that X can assume are the positive integers 1,2,3,4,5,6. Please give the corresponding probabilities for the values of X given below. Pr(X=1)= Pr(X = 2) = Pr(X = 3) = Pr(X = 4) = Pr(X = 5) = Pr(X = 6) = Further, find the probability that X is divisible by 2. Probability that X is divisible by 2 equals

11/36 9/36 7/36 5/36 3/36 1/36 15/36

x = 18,19,20,21,22 f(x)=8,9,11,8,4 Let x be the ages of students in a class. Given the frequency distribution F(x) above, determine the following probabilities: (a) P(x = 20) = (b) P(x > 19) = (c) P(18 ≤ x < 20) =

11/40 23/40 17/40

A bowl contains 9 red balls and 9 blue balls. A woman se- lects balls at random without looking at them. (a) How many balls must she select (minimum) to be sure of having at least three blue balls? (b) How many balls must she select (minimum) to be sure of having at least three balls of the same color?

12 5

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties: (a) are divisible by 7 (b) are divisible by 5 but not by 7. (c) are not divisible by either 5 or 7. (d) are divisible by 5 and by 7

1286 1543 6171 257

In a survey of 245 people, the following data were obtained relating gender to political orientation: See screenshot of table A person is randomly selected. What is the probability that the person is: a) Male? b) Male and a Democrat? c) Male given that the person is a Democrat? d) Republican given that the person is Male? e) Female given that the person is an Independent? f) Are the events Male and Democrat independent? Enter yes or no .

159/245 56/245 56/103 63/159 16/56 no

To examine the effectiveness of its four annual advertising promotions, a mail order company has sent a questionnaire to each of its customers, asking how many of the previous year's promotions prompted orders that would not have otherwise been made. The accompanying table lists the probabilities that were derived from the questionnaire, where X is the random variable representing the number of promotions that prompted orders. If we assume that overall customer behavior next year will be the same as last year, what is the expected number of promotions that each customer will take advantage of next year by ordering goods that otherwise would not be purchased? X=0,1,2,3,4 P(X)=.055,.248,.39,.175,.132 Expected value = A previous analysis of historical records found that the mean value of orders for promotional goods is 21 dollars, with the company earning a gross profit of 20% on each order. Calculate the expected value of the profit contribution next year. Expected value = The fixed cost of conducting the four promotions is estimated to be 18000 dollars with a variable cost of 4 dollars per customer for mailing and handling costs. What is the minimum number of customers required by the company in order to cover the cost of promotions? (Round your answer to the next highest integer.) Breakeven point =

2.081 8.7402 BreakEvenPoint = FixedCost / (ProfitPerPerson−VariableCostPerPerson) 3797

Employment data at a large company reveal that 57 % of the workers are married, that 45 % are college graduates, and that 1/2 of the college graduates are married. What is the probability that a randomly chosen worker is: a) neither married nor a college graduate? Answer = % b) married but not a college graduate? Answer = % c) married or a college graduate? Answer = %

20.5 34.5 79.5

Suppose that a department contains 9 men and 19 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

271320

The owner of a small firm has just purchased a personal com- puter, which she expects will serve her for the next two years. The owner has been told that she "must" buy a surge suppres- sor to provide protection for her new hardware against possible surges or variations in the electrical current, which have the ca- pacity to damage the computer. The amount of damage to the computer depends on the strength of the surge. It has been esti- mated that there is a 2% chance of incurring 500 dollar damage, 6% chance of incurring 200 dollar damage, and 12% chance of 75 dollar damage. An inexpensive suppressor, which would provide protection for only one surge, can be purchased. How much should the owner be willing to pay if she makes decisions on the basis of expected value? Expected value =

31

(1 point) A card is drawn at random from a regular playing card deck of 52 cards. Find the probability that a number card (A,1,2,3,4,5,6,7,8,9,10) is drawn. Answer:

40/52

In an experiment, a ball is drawn from an urn containing 13 blue balls and 10 green balls. If the ball is blue, a coin is flipped two times in succession. Otherwise a coin is flipped three times in succession. How many elements of the sample space will have a blue ball ? How many elements of the sample space are there altogether ?

52 132

How many different 10-letter words (real or imaginary) can be formed from the letters in the word ASSIGNMENT?

907200

Two events A and B are said to be independent if: • A. P(A and B) = P(A)+P(B) • B. P(A|B) = P(B) • C. P(B|A) = P(A) • D. P(A and B) = P(A)·P(B) If A and B are mutually exclusive events with P(A) = 0.70, then P(B): • A. can be any value between 0 and 1 • B. can be any value between 0 and 0.70 • C. cannot be larger than 0.30 • D. cannot be smaller than 0.30

D C