BINOMIALPROBABILITYDISTRIBUTION GEOMETRICPROBABILITYDISTRIBUTION

( ) 1n kkn

p x k p pk

Mean(Expectedvalue) =npVariance 2=np(1–p)StandardDeviation (1 )σ np p

RANDOMVARIABLESMean= i ix p x

StandardDeviation= 2

i ix p x

x=thenumberoftrialsuntilthefirstsuccessisobserved

p=probabilityof"success"onasingletrial 1

( ) 1x

p x p p

Mean(Expectedvalue) = 1

p

Variance 2=2

1 p

p

StandardDeviation =2

1 p

p

1. Among the voters in a certain precinct of Dallas. 80% are

Democrats. If ten voters are selected at random, determine the probability that:

a) exactly 5 are Democrats. b) exactly 7 are Democrats. c) at least 4 are Democrats. d) what is the expected number of Democrats in a

sample of size 25? e) what is the standard deviation for part (d)?

2. The following data are based on information taken from

the Statistical Abstract of the United States. In this table, x = size of family. The % data are the percentages of U.S. families of this size.

x 2 3 4 5 6 7 or more % 42% 23% 21% 10% 3% 1%

a) Convert the percentage data to probabilities and make a histogram of the probability distribution for family size.

b) What is the probability that a family selected at random will have only two members?

c) What is the probability that a family selected at random will have more than three members?

d) Compute , the expected family size (round families of size 7 or more to size 7).

e) Compute , the standard deviation (round families of size 7 or more to size 7).

3. Approximately 71% of all college students claim that the

main reason they are attending college is to make more money after graduation. In 1967, approximately 83% of all college students claimed they were attending college primarily to develop a meaningful philosophy of life. Let x be the random variable which represents the first college student selected at random who you encounter who says he or she is in college primarily to make more money. a) Write out the formula for the probability distribution

of the random variable. b) Find the probability that x = 1, x = 2, x 3

4. Approximately 3.6% of all (untreated) Jonathan apples have a bitter pit – a disease of apples resulting in a soggy core, which is caused either by over watering the apple tree or a calcium deficiency in the soil. Let n be a random variable that represents the first Jonathan apple chosen at random that has bitter pit. a) Find the probability that n = 3, n = 5. n = 12, and

n < 8. b) What is the probability that it takes at least 5 apples

before you select one which has bitter pit? 5. In a study of sleep patterns of adults, it has been found

that 23% of the sleep time is spent in the REM (rapid eye movement) stage. a) If a sleeping adult is observed 5 randomly selected

times, find the probability that exactly one of the five observations will be made during REM sleep.

b) Find the standard deviation for the number of REM stages observed in groups of five observations.

6. a. Find the mean, variance, and standard deviation of X.

X -1 0 1 2 P(X) 0.3 0.1 0.5 0.1

b. Find the mean, variance, and standard deviation of Y. Y 2 3 5

P(Y) 0.6 0.3 0.1 c) Let W = 3 + 2 X. Find the mean, variance, and

standard deviation of W. d) Let W = X + Y. Find the mean, variance, and

standard deviation of W. e) Let W = X – Y. Find the mean, variance, and

standard deviation of W. f) Let W = X + X. Find the mean, variance, and

standard deviation of W. g) Let W = 2X. Find the mean, variance, and standard

deviation of W. h) Let W = X – X. Find the mean, variance, and

standard deviation of W. i) Let W = 2X + 5Y. Find the mean, variance, and

standard deviation of W.

7. A stationery store has decided to accept a large shipment of ball-point pens if an inspection of 20 randomly selected pens yields no more than two defective pens. a) Find the probability that this shipment is accepted if

5% of the total shipment is defective. b) Find the probability that this shipment is not accepted

if 15% of the total shipment is defective. 8. On the leeward side of the island of Oahu in the small

village of Nanakuli, about 80% of the residents are of Hawaiian ancestry. a) What is the probability that the first person you meet

in Nanakuli is of Hawaiian descent? The second person? The third person?

b) What is the probability that you will meet at least 5 people before you meet someone who is of Hawaiian ancestry?

9. On Waikiki it is estimated that about 4% of the residents

are of Hawaiian ancestry. Repeat question #8 for Waikiki residents.

10. At Fontaine Lake Camp on Lake Athabasca in northern

Canada, history shows that about 30% of the guests catch lake trout over 20 pounds on 4-day fishing trip. Let n be a random variable that represents the first trip to Fontaine Lake camp on which a guest catches a lake trout over 20 pounds. a) Write out the formula for the probability distribution

of the random variable n. b) Find the probability that a guest catches a 20 pound

trout for the first time on trip number 3. On trip number 2.

c) Find the probability that it takes more than 4 trips for the guest to catch a 20 pound lake trout.

11. Police find that a patrol unit gets a 30% arrest record

when it sets up a checkpoint for drunk drivers. a) Find the probability that of 200 drivers checked, there

will be exactly one arrest. More than 3 arrests? b) What is the expected number of arrests? The standard

deviation of arrests? c) What is the probability that the third driver is the first

one arrested? That at least 10 drivers are checked before an arrest is made?

12. The Los Angeles Times (Dec. 13, 1992) reported that

what airline passengers like to do most on long flights is rest or sleep; in a survey of 3697 passengers, almost 80% did so. Suppose that for a particular route, the actual percentage is exactly 80%, and consider randomly selecting six passengers. Then x, the number among the selected six who rested or slept, is a binomial random variable with n = 6 and p = 0.8. a) Calculate P(4) b) Calculate P(6), the probability that all six selected

passengers rested or slept. c) Determine P(x 4).

13. Refer to #12, and suppose that ten rather than six passengers are selected (n = 10, p = 0.8). a) What is P(8)? b) Calculate P(x 7). c) Calculate the probability that more than half of the

selected passengers rested or slept. 14. Twenty-five percent of the customers entering a grocery

store between 5 P.M. and 7 P.M. use an express checkout. Consider five randomly selected customers, and let x denote the number among the five who use the express checkout. a) What is P(2), that is, P(x = 2)? b) What is P(x 1) c) What is P(2 x)? d) What is P(x 2)?

15. A breeder of show dogs is interested in the number of

female puppies in a litter. If a birth is equally likely to result in a male or female puppy, give the probability distribution of the variable

x = number of female puppies in a litter of size 5 16. Selected boxes of a breakfast cereal contain a prize.

Suppose that 5% of the boxes contain the prize and the other 95% contain the message "Sorry, try again." A consumer determined to find a prize decides to-continue to buy boxes of cereal until a prize is found. Consider the random variable x, where x = number of boxes purchased until a prize is found.

a. What is the probability that at most 2 boxes must be purchased?

b. What is the probability that exactly four boxes must be purchased?

c. What is the probability that more than four boxes must be purchased?

17. If the temperature in Florida falls below 32°F during

certain periods of the year, there is a chance that the citrus crop will be damaged. Suppose that the probability is 0.1 that any given tree will show measurable damage when the temperature falls to 30°F. If the temperature does drop to 30°F, what is the expected number of trees showing damage in orchards of 2000 trees? What is the standard deviation of the number of trees that show damage?

18. Thirty percent of all automobiles undergoing an emission

inspection at a certain inspection station fail the inspection. a) Among 15 randomly selected cars, what is the

probability that at most 5 fail the inspection? b) Among 15 randomly selected cars, what is the

probability that between 5 and 10 (inclusive) fail to pass inspection?

c) Among 25 randomly selected cars, what is the mean value of the number that pass inspection, and what is the standard deviation of the number that pass inspection?

19. You are to take a multiple-choice exam consisting of 100 questions with five possible responses to each. Suppose that you have not studied and so must guess (select one of the five answers in a completely random fashion) on each question. Let x represent the number of correct responses on the test. a) What kind of probability distribution does x have? b) What is your expected score on the exam? c) Compute the variance and standard deviation of x.

20. Which of the following are continuous variables, and

which are discrete? a) Number of traffic fatalities per year in the state of

Florida b) Distance a golf ball travels after being hit with a

driver c) Time required to drive from home to college on any

given day d) Number of ships in Pearl Harbor on any given day e) Your weight before breakfast each morning f) Cost of an adult ticket at each of the movie theaters in

your town 21. Which of the following are continuous variables, and

which are discrete? a) Amount of sleep you got last night b) Home team score in a basketball game c) Number of ducks sitting on a pond d) BTUs absorbed by a solar panel e) Volume of water in Lake Powell f) Number of prisoners in the county jail

22. The National Hockey League keeps a day-to-day total of

all goals scored in the league. Games consist of three periods. An overtime (OT) period is played in the event that there is a tied score at the end of the third period. Through March 20, 1990, there was a total of 5747 goals scored in the NHL during the 1989-1990 season. The breakdown by periods is as follows:

Period x 1 2 3 OT Goals Scored f 1776 2035 1883 53

a) A scoring play is chosen at random from games played in the 1989-1990 NHL season up through March 20, 1990. Use relative frequencies to calculate the probability P(x) that the goal was made in the x = 1st, 2nd. 3rd, OT period

b) Use a histogram to graph the probability distribution. Note: Consider OT as a 4th period with x = 4.

c) Find the expected value of the distribution. How can you interpret this value?

d) Find the standard deviation of the distribution. e) In which period is a goal most likely to be scored? If

you picked a scoring play at random from the games included in the table above, what is the probability that the goal was not scored in the second period of a game?

23. It is found that for one section of Interstate 30, 94% of the vehicles are traveling at speeds greater than 65 mph. Find the probability that among 24 randomly selected vehicles, less than 20 are traveling above 65 mph.

24. The head nurse on the third floor of a community hospital

is interested in the number of nighttime room calls requiring a nurse. For a random sample of 208 nights (9:00 P.M. to 6:00 A.M.), the following information was obtained, where x = number of room calls requiring a nurse and f = frequency with which this many calls occurred (i.e., number of nights). x 36 37 38 39 40 41 42 43 44 45 f 6 10 11 20 26 32 34 28 25 16

a) If a night is chosen at random from these 208 nights, use relative frequencies to find P(x) when x = 36, 37, 38, 39, 40, 41, 42, 43, 44, and 45.

b) Use a histogram to graph the probability distribution of part a.

c) Assuming these 208 nights represent the population of all nights at community hospital, what do you estimate the probability is that, on a randomly selected night, there will be from 39 to 43 (including 39 and 43) room calls requiring a nurse?

d) What do you estimate the probability is that there will be from 36 to 40 (including 36 and 40) room calls requiring a nurse?

e) Find the expected number of room calls requiring a nurse.

f) Find the standard deviation of the x distribution. 25. The following data are based on information taken from

Daily Creel Summary (Feb. 28, 1993), published by the Paiute Indian Nation, Pyramid Lake, Nevada. Movie stars and U.S. presidents have fished Pyramid Lake. It is one of the best places in the lower 48 states to catch trophy cutthroat trout. In this table, x = number of fish caught in a 6-hour period. The % data are the percentages of fishermen who caught x fish in a 6-hour period while fishing from shore.

x 0 1 2 3 4 or more % 44% 36% 15% 4% 1%

a) Convert the percentages to probabilities and make a histogram of the probability distribution.

b) Find the probability that a fisherman selected at random fishing from shore catches one or more fish in a 6-hour period.

c) Find the probability that a fisherman selected at random fishing from shore catches two or more fish in a 6-hour period.

d) Compute , the expected value of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4).

e) Compute , the standard deviation of the number of fish caught per fisherman in a 6-hour period (round 4 or more to 4).

26. Almost all new independent businesses (not franchises) that fail do so in the first 10 years. The following data are based on information from the Statistical Abstract of the United States (112th ed). For the population of new businesses that fail, let x = the year in which the business fails. For instance, x = 3 means the business fails in its third year. The % data are the proportions of such businesses that fail during the xth year of the business. x % a) Make a histogram of the probability

distribution. b) Find the probability that a business

selected at random that fails will fail in its third year or before.

c) Find the probability that a business selected at random that fails will fail in its fifth year or later.

d) Compute , the expected value for the year in which a business that fails will fail,

e) Compute , the standard deviation for the year in which a business that fails will fail.

1 0.02 2 0.07 3 0.15 4 0.18 5 0.21 6 0.16 7 0.10 8 0.06 9 0.04 10 0.01

27. USA Today (June 2, 1993) reported that approximately

24% of all state prison inmates released on parole become repeat offenders. Suppose the parole board is examining five prisoners up for parole. Let x = number of prisoners out of five on parole who become repeat offenders.

x 0 1 2 3 4 5 P(x) 0.254 0.400 0.253 0.080 0.012 0.001 a) Find the probability that one or more of the five

parolees will be repeat offenders. How does this number relate to the probability that none of the parolees will be repeat offenders?

b) Find the probability that two or more of the five parolees will be repeat offenders.

c) Find the probability that four or more of the five parolees will be repeat offenders.

d) Compute , the expected number of repeat offenders out of five.

e) Compute , the standard deviation of the number of repeat offenders out of five.

28. Mr. Dithers wants to insure his yacht for $80,000. The

Big Rock Insurance Company estimates a total loss may occur with a probability of 0.005, a 50% loss with probability of 0.01, and a 25% loss with probability 0.05. If Big Rock will pay no benefits for any other partial loss, what premium should Mr. Dithers pay each year if Big Rock wants to make a profit of $250?

29. The student senate is sponsoring a car raffle to buy playground equipment for disadvantaged children. The senate buys a used car for $3000 and sells 3750 raffle tickets at $1.50 per ticket. a) If you buy 30 tickets, what is the probability that you

will win the car? What is the probability that you will not win the car?

b) Your expected earnings can be found by multiplying the value of the car by the probability that you will win it. What are your expected earnings? Is it more or less than the amount you paid for 30 tickets? How much did you effectively contribute to buy playground equipment?

30. Sophie is a dog who loves to play catch. Unfortunately,

she isn't very good, and the probability that she catches a ball is only 0.1. Let x = number of tosses required until Sophie catches a ball. a) Does x have a binomial or a geometric distribution? b) What is the probability that it will take exactly two

tosses for Sophie to catch a ball? c) What is the probability that more than three tosses

will be required? 31. In a study of middle-aged adults (40 to 65 years), it is

found that 7.8% suffer from hypertension. A follow-up study begins with a random selection of 40 middle-aged adults. a) Find the probability that exactly one-fourth of the

sample suffers from hypertension. b) Find the mean number of hypertension cases found in

such groups of 40. c) Find the standard deviation for the numbers of

hypertension cases in groups of 40. d) What is the probability that less than 4 cases of

hypertension will be found in a group of 40. e) What is the probability that at least 3 cases of

hypertension will be found in a group of 40.

Answers 1. a. 0.0264 b. 0.201 c. 0.9991 d. 20 e. 2 2. a.

b. 0.42 c. 0.35 d. 3.12 e. 1.202 3. a. p(x) = .71(.29 )x-1 b. p(x = 1) = .71(.29)1-1 = 0.71 p(x = 2) = .71(.29)2-1 = 0.2059 p(x 3) = 1- p(x = 1) - p(x = 2) = 0.0841 4. a. p(n = 3) = .036(:964)3-1 = 0.0335 p(n = 5) = .036(.964)5-1 = 0.0311 p(n = 12) = .036(:964)12-1 = 0.0241 p(n < 8) = 0.2264 b. 0.8636 5. a. .4043 b. .941 6. a. = 0.4; = 1.02; 2 = 1.04 b. = 2.6; = 0.917; 2 = 0.84 c. = 3.8; = 2.04; 2 = 4.16 d. = 3; = 1.37; 2 = 1.88 e. = -2.2; = 1.37; 2 = 1.88 f. = 0.8; = 1.44; 2 = 2.08 g. = 0.8; = 2.04; 2 = 4.16 h. = 0; = 1.44; 2 = 2.08 i. = 12.2; = 5.016; 2 = 25.16 7. a. 0.9245 b. 0.5951 8. a. 0.8; 0.16; 0.032 b. 0.00032 9. a. 0.04; 0.0384; 0.036864 b. 0.815 10. a. p(n) = 0.3(0.7)n-1 b. 0.147; 0.21 c. 0.2401 11. a. < 0.0001; > 0.9999 b. = 60; = 6.481 c. 0.147; 0.0404 12. a. 0.24576 b. 0.262 c. 0.9011

13. a. 0.302 b. 0.322 c. 0.966 14. a. 0.264 b. 0.633 c. 0.367 d. 0.736 15.

x p(x) 0 0.03125 1 0.15625 2 0.3125 3 0.3125 4 0.15625 5 0.03125

16. a. 0.0975 b. 0.0429 c. 0.8145 17. = 200, = 13.416 18. a. 0.722 b. 0.484 c.7.5, 2.2913 19. a. Binomial b. 20 c. 16; 4 20. (a) Discrete (b) Continuous (c) Continuous (d) Discrete (e) Continuous (f) Discrete 21. (a) Continuous (b) Discrete (c) Discrete (d) Continuous (e) Continuous (f) Discrete 22. (a)

x 1st 2nd 3rd OT P(x) 0.309 0.354 0.328 0.009

(b) Goals by Game Period for NHL 89-90

23. 0.0127

42%

23% 21%

10%3% 1%

0%10%20%30%40%50%

2 3 4 5 6 7 ormore

Per

cen

t

US Family Size

0

0.1

0.2

0.3

0.4

1 2 3 4 (OT)

24. a. and b.

c. 0.673 d. 0.351 e. 41.288 f. 2.326 25. a.

b. 0.56 c. 0.20 d. 0.82 e. 0.899 26. a.

b. 0.34 c. 0.58 d. 4.618 e. 2.166

27. a. 0.746; 1 – 0.254 b. 0.346 c. 0.013 d. = 1.199 e. = 0.955 28.

x P(x) -80000 0.005 -40000 0.01 -20000 0.05

0 0.935 = 1800 + 250 = 2050 29. a. 30/3750; 3720/3750 b. 3000*30/3750 = 24; less; 45-24=21 30. a. geometric b. 0.09 c. 0.729 31. a. 0.0006 b. 3.12 c. 1.696 d. 0.6194 e. 0.6129

44%36%

15%

4% 1%0%

10%

20%

30%

40%

50%

0 1 2 3 4 ormore

Per

cen

t

Number of Fish