Discrete RVsx# days x cars soldf(x)054111727234241253Total

DiCarlo Motors in Saratoga New York has tracked the sale of their automobiles over the past 300 days. They wish to use this data to estimate their expected sales on any given day.a. use the relative frequency approach to construct a probability distribution for the number of cars sold in a given dayb. draw a graph of the probability distribution.c. show that your probability distribution satisfies the required conditions for a valid discrete probability distribution

Discrete RVs.sx# days x cars soldf(x)0540.1811170.392720.243420.144120.04530.01Total3001

Discrete RVs.s

f(x)

Exercise 7xf(x)200.2250.15300.25350.4Total11. Is this distribution valid?2. What is the probability that x = 30?3. What is the probability that x is less than or equal to 25?4. What is the probability that x is greater than 30?

DiCarlo Motors in Saratoga New York has tracked the sale of their automobiles over the past 300 days. They wish to use this data to estimate their expected sales on any given day.a. use the relative frequency approach to construct a probability distribution for the number of cars sold in a given dayb. draw a graph of the probability distribution.c. show that your probability distribution satisfies the required conditions for a valid discrete probability distribution

Exercise 7.sxf(x)Cumulative Distribution200.20.2250.150.35300.250.6350.41Total1Solutions:1. Is this distribution valid?Yes, because the probabilities are greater than or equal to 0 and 1 individually, and also sum to one overall2. What is the probability that x = 30?0.253. What is the probability that x is less than or equal to 25?0.354. What is the probability that x is greater than 30?0.4

Exercise 9AgeNumber of Childrenf(x)x * f(x)(x - E(x))^2 * f(x)637,369787,4368160,8409239,71910286,71911306,53312310,78713302,60414289,168Total

Expected value of xVariance of xExpected value of x using the SumProduct formulaNationally, 38% of fourth graders cannot read an age-appropriate book. The following data show the number of children, by age identified as learning disabled under special education. Most of these children have reading problems that should be identified and corrected before third grade. Current federal law prohibits most children from receiving extra help from special education programs until they fall behind by approximately two years' worth of learning, and that typically means third grade or later (USA Today, September 6, 2001).

Suppose that we wanted to select a sample of children identified as learning disabled under special education for a special program designed to improve reading ability. Let x be a random variable indicating the age of one randomly selected child.

a. Use the data to develop a probability distribution for x. Specify the values for x and the corresponding values for the probability function f(x) b. Draw a graph of the probability distribution. x. c. Show that the probability distribution satisfied the requirements of a discrete random variable.d. Calculate the expected value and variance of x.

Exercise 9.sAgeNumber of Childrenf(x)x * f(x)(x - E(x))^2 * f(x)637,3690.0180.11093250210.4620571362787,4360.0430.30281989440.69185723348160,8400.0800.6366197880.71577990169239,7190.1191.06743404210.474000465910286,7190.1421.41857582840.14160965511306,5330.1521.66826870510.000000115912310,7870.1541.84518609220.154034482313302,6040.1501.94631934390.599391161114289,1680.1432.00296955981.2883738624Total2,021,1751.00010.99912575614.527104013910.9991257561

Nationally, 38% of fourth graders cannot read an age-appropriate book. The following data show the number of children, by age identified as learning disabled under special education. Most of these children have reading problems that should be identified and corrected before third grade. Current federal law prohibits most children from receiving extra help from special education programs until they fall behind by approximately two years' worth of learning, and that typically means third grade or later (USA Today, September 6, 2001).

Suppose that we wanted to select a sample of children identified as learning disabled under special education for a special program designed to improve reading ability. Let x be a random variable indicating the age of one randomly selected child.

a. Use the data to develop a probability distribution for x. Specify the values for x and the corresponding values for the probability function f(x) b. Draw a graph of the probability distribution. x. c. Show that the probability distribution satisfied the requirements of a discrete random variable.d. Calculate the expected value and variance of x.

Exercise 9.s

f(x)

Binomial# trials (n)3# successes in n trial =2P(success) =0.3P(x)0.189possible outcomespdfcdfxf(x)F(x)(x-E(x))^20123TotalExpected Value of xVariance of x

Expected value of xVariance of xExpected value of x using the SumProduct formulaMartin Clothing Store Problem:Let us consider the purchase decisions of the next three customers who enter the store. On the basis of past experience, the store manager estimates the probability that any one customer will make a purchase is .30.

a. What is the probability that exactly two of the next three customers will make a purchase?

b. Construct a bar graph of the pdf and cdf for this problem.

c. What is the probability that at least two customers make a purchase?

d. How many purchases should the store managers expect to make over the next three customers?

e. What is the variance of the customers purchasing behavior?

Binomial.s# trials (n)3# successes in n trial =2P(success) =0.3P(x)0.189possible outcomespdfcdfxf(x)F(x)(x-E(x))^2X*f(X)00.3430.3430.81010.4410.7840.010.44120.1890.9731.210.37830.02714.410.081Total1Expected Value of x0.90.9Variance of x0.630.63

Martin Clothing Store Problem:Let us consider the purchase decisions of the next three customers who enter the store. On the basis of past experience, the store manager estimates the probability that any one customer will make a purchase is .30.

a. What is the probability that exactly two of the next three customers will make a purchase?

b. Construct a bar graph of the pdf and cdf for this problem.

c. What is the probability that at least two customers make a purchase?

d. How many purchases should the store managers expect to make over the next three customers?

e. What is the variance of the customers purchasing behavior?

Binomial.s

f(x)

Exercise 28

F(x)

Exercise 28.sa)P(X=2)b)P(X>=2)c)P(X=0)

The 1999 Youth and Money Survey, sponsored by the American Savings and Education Council, the Employee Benefit and Research Institute, and Matthew Greenwald & Associates, talked to 1000 students ages 16-22 about personal finance. The survey found that 33% of the students have their own credit card.

a. In a sample of 6 students, what is the probability that two will have their own credit card?

b. In a sample of 6 students, what is the probability that at least two will have their own credit card?

c. In a sample of 10 students, what is the probability that none will have their own credit card.

Poisson1a)P(X=2)0.3291685615b)P(X>=2)0.6422168466c)P(X=0)0.018228378

The 1999 Youth and Money Survey, sponsored by the American Savings and Education Council, the Employee Benefit and Research Institute, and Matthew Greenwald & Associates, talked to 1000 students ages 16-22 about personal finance. The survey found that 33% of the students have their own credit card.

a. In a sample of 6 students, what is the probability that two will have their own credit card?

b. In a sample of 6 students, what is the probability that at least two will have their own credit card?

c. In a sample of 10 students, what is the probability that none will have their own credit card.

Poisson1.smean =10xf(x)F(x)0using pdfusing cdf1A2B3C4D5678910111213141516171819202122232425

Suppose that we are interested in the number of arrivals at the drive-up teller window of a bank during a 15 minute period on the weekday mornings. Historical data shows that the average arrival rate is 10 cars per 15 minute intervals.

A. What is the probability that exactly 4 cars will arrive in the next 15 minute interval?

B. What is the probability that more than 12 cars will arrive in the next 15 minute interval?

C. What is the probability that at least 7 cars but less than 15 cars will arrive in the next 15 minute interval?

D. What is the probability that more than two cars will arrive in the next 15 minute interval?

Poisson2mean =10xf(x)F(x)00.00004539990.0000453999using pdfusing cdf10.00045399930.0004993992A0.01891660.018916620.00226999650.0027693957B0.20842580.208443530.0075666550.0103360507C0.78640010.786400140.01891663740.0292526881D0.99721290.997230650.03783327480.067085962960.0630554580.1301414209Alternative70.09007922570.22022064660.018916680.11259903210.33281967880.208443590.12511003570.45792971450.7864001100.12511003570.58303975020.9972306110.11373639610.6967761463120.09478033010.7915564764130.07290794620.8644644226140.05207710440.9165415271150.03471806960.9512595967160.02169879350.9729583902170.01276399620.9857223864180.0070911090.9928134954190.00373216260.996545658200.00186608130.9984117393210.00088861010.9993003495220.00040391370.9997042632230.00017561470.9998798778240.00007317280.9999530506250.00002926910.9999823197

Suppose that we are interested in the number of arrivals at the drive-up teller window of a bank during a 15 minute period on the weekday mornings. Historical data shows that the average arrival rate is 10 cars per 15 minute intervals.

A. What is the probability that exactly 4 cars will arrive in the next 15 minute interval?

B. What is the probability that more than 12 cars will arrive in the next 15 minute interval?

C. What is the probability that at least 7 cars but less than 15 cars will arrive in the next 15 minute interval?

D. What is the probability that more than two cars will arrive in the next 15 minute interval?

Poisson2.smean6xf(x)0A.1B.234567891011121314151617181920Total

Suppose that we learn that major defects on a stretch of highway, one month after resurfacing, occur at the average rate of two per mile.

A. What is the probability that no major defects in a 3 mile stretch of highway exists?

B. What is the probability that at least one major defect exists in the three mile section?

Exercise 42mean6xf(x)00.0024787522A.0.002478752210.0148725131B.0.997519792720.044617539230.089235078440.133852617550.16062314160.16062314170.13767697880.103257733590.068838489100.0413030934110.02252896120.01126448130.0051989908140.0022281389150.0008912556160.0003342208170.0001179603180.0000393201190.0000124169200.0000037251Total0.9999985449

Suppose that we learn that major defects on a stretch of highway, one month after resurfacing, occur at the average rate of two per mile.

A. What is the probability that no major defects in a 3 mile stretch of highway exists?

B. What is the probability that at least one major defect exists in the three mile section?

Exercise 42.s1 minute30 secondmean73.5xf(x)f(x)01a.2b.3c.4d.567891011121314151617181920Sum

More that 50 million guests stayed at bed and breakfasts (B&Bs) last year. The web site for B&Bs Inns North America, which averages approximately seven visitors per minute enables many B&Bs to attract guests without waiting years to be mentioned in guide books (Time, September 2001).

a. Compute the probability of no web site visitors in a 1 minute period.

b. Compute the probability of two or more web site visitors in a 1 minute period.

c. Compute the probability of one or more web site visitors in a 30 second period.

d. Compute the probability of five or more web site visitors in a 1 minute period.

Hypergeometric 11 minute30 secondmean73.5xf(x)f(x)00.0009118820.030197383410.00638317380.105690842a.0.00091188220.02234110820.1849589735b.0.992690448930.05212925240.215785469c.0.969802616440.09122619160.1888122854d.0.827008392150.12771666830.132168599860.14900277970.077098349970.14900277970.038549174980.13037743220.01686526490.10140466950.0065587138100.07098326870.0022955498110.0451711710.0007304022120.02634984970.000213034130.01418838060.0000573553140.00709419030.0000143388150.00331062210.0000033457160.00144839720.0000007319170.00059639880.0000001507180.00023193290.0000000293190.0000854490.0000000054200.00002990710.0000000009Sum0.99998550470.9999999998

More that 50 million guests stayed at bed and breakfasts (B&Bs) last year. The web site for B&Bs Inns North America, which averages approximately seven visitors per minute enables many B&Bs to attract guests without waiting years to be mentioned in guide books (Time, September 2001).

a. Compute the probability of no web site visitors in a 1 minute period.

b. Compute the probability of two or more web site visitors in a 1 minute period.

c. Compute the probability of one or more web site visitors in a 30 second period.

d. Compute the probability of five or more web site visitors in a 1 minute period.

Hypergeometric1.sa.b.N55n23r33x22Answer:

a. Suppose that we wish to select a committee of two people taken from a population of 5 people. The five member population consists of 3 women and 2 men. If we randomly select two people, what is the probability that the two people are both women?

b. Suppose that three members are to be selected, what is the probability that two are women and one is a man?

Hypergeometric2a.b.N55n23r33x22Answer:0.30.6N5n3r2x1Answer:0.6Person APerson BPerson CSexProbSexProbSexProbP(A)*P(B)*P(C )Suc/FailM0.4M0.25M00FailM0.4M0.25F10.1FailM0.4F0.75M0.33333333330.1FailSum of Successes:0.6M0.4F0.75F0.66666666670.2SuccessF0.6M0.5M0.33333333330.1FailF0.6F0.5M0.66666666670.2SuccessF0.6M0.5F0.66666666670.2SuccessF0.6F0.5F0.33333333330.1Fail

a. Suppose that we wish to select a committee of two people taken from a population of 5 people. The five member population consists of 3 women and 2 men. If we randomly select two people, what is the probability that the two people are both women?

b. Suppose that three members are to be selected, what is the probability that two are women and one is a man?Note on part B.Obtaining 2 woman and one man, in a sample of 3 people is equivalent with obtaining:

Only two women in a sample of 3 people (thus, the third person will be a man) orOnly one man in a sample of 3 people (thus, the other two people will be women).

Hypergeometric2.sN10n3r4x2Answer:

Suppose that a population consists of 10 items, four of which are classified defective and six which are classified as non-defective. What is the probability that a random selection of three will provide two defective items.

Hypergeometric 3N10n3r4x2Answer:0.3

Suppose that a population consists of 10 items, four of which are classified defective and six which are classified as non-defective. What is the probability that a random selection of three will provide two defective items.

Hypergeometric3.sa.b.c.N525252n222r20416x222dAnswer:

ABBlackjack, or twenty-one as it frequently is called, is a popular gambling game played in Las Vegas casinos. A player is dealt two cards. Face cards (jacks, queens, kings) and tens have a point value of 10. Aces have a point value of 1 or 11. A 52 card deck has 16 cards with a point value of 10 (jacks, queens, kings, tens) and four aces.

a. What is the probability that both cards dealt (on the first hand) are aces or ten point cards?

b. What is the probability that both cards are aces?

c. What is the probability that both cards are ten point cards?

d. What is the probability of getting 21 on the first hand?

Hypergeometric 4a.b.c.N525252n222r20416x222dAnswer:0.14328808450.00452488690.09049773760.04826546A = P(exactly one ace on the first draw of two sequential draws )B = P(exactly one card worth 10 points on the second draw of two sequential draws)A intersect B = P(getting 21)ABN5251n11r416x11P(success)0.07692307690.31372549020.02413273Vice versaABN5251n11r164x11P(success)0.30769230770.07843137250.02413273

Blackjack, or twenty-one as it frequently is called, is a popular gambling game played in Las Vegas casinos. A player is dealt two cards. Face cards (jacks, queens, kings) and tens have a point value of 10. Aces have a point value of 1 or 11. A 52 card deck has 16 cards with a point value of 10 (jacks, queens, kings, tens) and four aces.

a. What is the probability that both cards dealt (on the first hand) are aces or ten point cards?

b. What is the probability that both cards are aces?

c. What is the probability that both cards are ten point cards?

d. What is the probability of getting 21 on the first hand?AB

Hypergeometric 4.sN1010101010n34567r22222x11111P(1 defect)N1010101010n34567r22222x22222P(2 defects)a.b.c.P(1 or 2 defects)Answers:abcd

A shipment of 10 items has two defective and eight non-defective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected.

a. if a sample of 3 items is selected, what is the probability that the shipment will be rejected?

b. if a sample of 4 items is selected, what is the probability that the shipment will be rejected?

c. if a sample of 5 items is selected, what is the probability that the shipment will be rejected?

d. If management would like a .9 probability of rejecting a shipment with two defective items out of 10 total items, what sample size would you recommend?

N1010101010101010n345678910r22222222x11111111P(1 defect)0.46666666670.53333333330.55555555560.53333333330.46666666670.35555555560.20N1010101010101010n345678910r22222222x22222222P(2 defects)0.06666666670.13333333330.22222222220.33333333330.46666666670.62222222220.81a.b.c.P(1 or 2 defects)0.53333333330.66666666670.77777777780.86666666670.93333333330.977777777810Answers:a0.5333333333b0.6666666667c0.7777777778d7

A shipment of 10 items has two defective and eight non-defective items. In the inspection of the shipment, a sample of items will be selected and tested. If a defective item is found, the shipment of 10 items will be rejected.

a. if a sample of 3 items is selected, what is the probability that the shipment will be rejected?

b. if a sample of 4 items is selected, what is the probability that the shipment will be rejected?

c. if a sample of 5 items is selected, what is the probability that the shipment will be rejected?

d. If management would like a .9 probability of rejecting a shipment with two defective items out of 10 total items, what sample size would you recommend?