Probability Questions and Answers

An experiment is pulling a ball from an urn that contains 8 blue balls and 8 red balls a Find the probability of getting a blue ball Leave your answer as a fraction b Find the probability of getting a red ball Leave your answer as a fraction

Rewrite this number in appropriate scientific notation 0 000141

Which means the same as 5 1411 x 105 514110 O 5141 1 5141100 O 51411

Write the decimal that means the same as 3 321 x 105

A piece of paper has a thickness of 4 2 x 10 inch Which decimal is equivalent to 4 2 10 0 00042 0 042 00 42 0 0042

The planet Mercury is approximately 36 000 000 miles from the sun Which is the distance written in scientific notation 03 6 106 36 x 107 36 x 106 03 6 107

Rewrite this number in appropriate scientific notation 328 000 answer

Rewrite this number as a whole number or decimal 6 7 x 105 answer

ewsletter publisher believes that 63 of their readers own a Rolls Royce A testing firm believes this is inaccurate and performs a test to dispute the publisher s m After performing a test at the 0 10 level of significance the testing firm fails to reject the null hypothesis at is the conclusion regarding the publisher s claim swer O There is sufficient evidence at the 0 10 level of significance that the percentage is not 63 O There is not sufficient evidence at the 0 10 level of significance that the percentage is not 63 Tables Keyp Keyboard Shor

7 From past experience it is known 90 of 1 year old children can distinguish their mother s voice from the voice of a similar sounding female A random sample of 20 1 year olds is given this voice recognition test 12 points HINT This is a binomial distribution n p q HINT 2 The value of p will change depending on the children recognizing or the children not recognizing the voice A Find the probability at least 3 children do not recognize their mother s voice B Find the probability all 20 children recognize their mother s voice I C Find the mean and standard deviation of x where x is the random variable denoting the number of children who recognize their mother s voice D Find the probability that at most 4 children recognize their mother s voice

I 1 75 Multiple Choice 3 points each The expected number of 1 s or 2 s in 240 rolls of a six sided die is a 60 a 50 15 b 40 2 The standard deviation of a binomial distribution for which n 50 and p 0 15 is a 0 599 b 7 082 a 2 5 c 80 b 0 914 3 If the random variable x is binomially distributed with n 10 and p 0 05 then P x 2 is c 6 375 b 0 0821 d 120 c 0 074 d 2 525 4 Given a Poisson random variable x where the average number of times an event occured in a certain period of time is 2 5 then P x 0 is c 1 5811 d 0 55 d 0 40 Which of the following experiments can be modeled by the Poisson distribution a The number of calls received by a switchboard in 1 hour b The number of bacteria per 22 ml of fluid c The number of customer arrivals at a checkout counter in 1 day d All of the above

12 In a food processing and packaging plant there are on the average two packaging machine breakdowns per week Assume the weekly machine breakdowns follow a Poisson distribution 10 points A Find the mean and standard of this distribution B Find the probability that there are no machine breakdowns in a given week C Find the probability that there are no more than two machine breakdowns in a given week I D Find the probability that there are 3 or more machine breakdowns in a given week

1 75 6 Let x denote the number of cars sold at Hatton Ford during a given week and p x is the probability that x occurs Complete the table to answer the questions that follow 11 points X 0 1 234 5 TOTALS p x 06 12 20 25 30 07 x p x x p x A Find the average number of cars sold in any given week B Find the variance and the standard deviation of the number of cars sold C Find each listed probability 1 P x 4 2 P 0 x 3 3 P 0 x 5

D Find the probability that at most 4 children recognize their mother s voice 8 Consider an experiment with 25 trials where the probability of success on any trial is 0 04 and let the random variable x be the number of successes among the 25 trials Find each probability listed below 11 points A P x 0 B P x 1 C P x 2 D P x 3 X E Explain how you could use the answers from A D to determine P x 4 p q n

9 The number of telephone calls coming into a business s switchboard averages 4 calls per minute Let x be the number of calls received 10 points A This is a B Find P x 0 distribution C Find the probability there will be at least 1 call a given 1 minute period D Find the probability there will be exactly 4 calls in a given 1 minute period E Calculate the mean and standard deviation for x the number of call received per minute at this switchboard

11 From a group of 10 bank officers 3 are selected to be relocated and supervise new branch offices If two of the 10 officers are women and 8 are men what is the probability exactly one of the officers to be relocated will be a woman 6 points A This is a B r b n distribution C Answer the question posed above X

13 A new surgical procedure is said to be successful 80 of the time Suppose the operation is performed five times and the results are assumed to be independent of one another 12 points A List the values A may take on B p q n C What is the probability that all 5 operations are successful D What is the probability exactly 4 are successful E What is the probability that less than 2 are successful A F If less than 2 operations were successful how would you feel about the performance of the surgical team Explain

10 Suppose that the probability that an adult in America will watch the upcoming World Series is 40 Each person is considered independent We are interested in the number of adults in America we must survey until we find one who will watch the World Series 10 points A This is a B List the values that X may take on C How many adults in America do you expect to survey until you find one who will watch the World Series distribution p VD Find the probability that you must ask 7 people I E Find the probability that you must ask 3 or 4 people

performance of the sur BONUS A package of 6 light bulbs contains 2 defective bulbs If 3 bulbs are selected for use find the probability none are defective 4 points

6 Assume each outcome in the sample space represented by the Venn diagram below is equally likely Find P AB A B Round your answer to the nearest hundredths Answer Previous Answer 0 79 Return to queniqu

Question 7 of 15 7 Find the variance of the sample data set 3 4 6 5 2 Answer Previous Answer 2 5

The table shows the educational attainment of the population of a country ages 25 and over expressed in millions Find the probability that a randomly selected person aged 25 or over has completed four years of high school only or some college but less than four years Male Female Total Years of High School Less than 4 14 4 only 31 15 29 The probability is Type an integer or a simplified fraction 28 59 Years of College Some less than 4 22 20 42 4 or more 24 16 40 Total 79 170

The game of Scrabble has 100 tiles The diagram shows the number of tiles for each letter and the letter s point value One tile is drawn from Scrabble s 100 tiles Find each probability a Find the probability of selecting a V b Find the probability of not selecting a V a The probability of selecting a V is Type an integer or a simplified fraction A 9 F2 K P 2 U4Z B 2 G 3 L 4 Q V 2 C2 H2 M2 R6 W2 D 4 N 6 S X 1 O T Y 2 E 12 1 9 J 2A Paint

The table shows the educational attainment of the population of a certain country ages 25 and over expressed in millions Find the probability that a randomly selected person aged 25 or over has completed four years of high school only or is male The probability is Male Female Total Years of High School Less than 4 16 14 30 4 only 29 32 61 Type an integer or a simplified fraction Years of College Some less than 4 25 3 25 50 4 or more 19 16 35 Total 89 87 176

The recessive alleles for cinnibar eyes cn and vestigial wings vg identify two autosomal genes on the third chromosome of Drosophila melanogaster When females heterozygous at these two genes cn cn vg vg were crossed with cinnibar eyed vestigial winged males the following classes and numbers of progeny n 1000 were obtained Wild type eye color wild type wings 42 Wild type eye color vestigial wings 464 Cinnibar eye color vestigial wings 38 Cinnibar eye color wild type wings 456 Based upon these results the map distance between the cn and vg genes is estimated to be 4 map units 8 map units 46 map units 92 map units

4 A college president wants to find out which courses students consider to be of the most benefit to them Which procedure would be most likely to produce a statistically unbiased sample a asking students to mail in a questionnaire b surveying a random sample of students taken from the list of all students C surveying the first hundred students on an alphabetical list d having students complete a questionnaire on the college web site

2 Which method is most likely to produce a random sample of the members of your class a listing the first six students that come to mind b choosing the five oldest students in the class writing the name of each student on a separate piece of paper and then drawing these slips from a hat d selecting the first six students to arrive at class

1 What is the best sampling technique to use for determining the average speed of the cars o section of highway a simple random sample b systematic sample c d convenience sample a or b

List all the permutations of five objects x y z s and t taken two at a time without repetition What is 5P2 List all the permutations of five objects x y z s and t taken two at a time without repetition Choose the correct answer below OA x y z s t O B xx xy xz xs xt yy yz ys yt zz zs zt ss st tt OC xy xz xs xt yz ys yt zs zt st O D xy xz xs xt yx yz ys yt zx zy zs zt sx sy sz st tx ty tz ts What is 5P2 Simplify your answer

1 Evaluate the expression below given the data set 2 4 6 8 2x 3 Answer Previous Answer 28 Return to overview

List all the combinations of five objects a b c d and e taken two at a time What is 5C2 List all the combinations of five objects a b c d and e taken two at a time Choose the correct answer below OA a b c d e B aa ab ac ad ae bb bc bd be cc cd ce dd de ee C ab ac ad ae ba bc bd be ca cb cd ce da db dc de ea eb ec ed O D ab ac ad ae bc bd be cd ce de What is 5C2 Cimplify your onewer

A lottery exists where balls numbered 1 to 24 are placed in an urn To win you must match the five balls chosen in the correct order How many possible outcomes are there for this game The number of possible outcomes is Simplif

Suppose that E and F are two events and that P E and F 0 1 and P E 0 4 What is P FIE P FIE Type an integer or a decimal

Suppose Emerson loses 30 of all checker games a What is the probability that Emerson loses two checker games in a row b What is the probability that Emerson loses three checker games in a row c When events are independent their complements are independent as well Use this result to determine the probability that Emerson loses three checker games in a row but does not lose four in a row a The probability that Emerson loses two checker games in a row is Round to four decimal places as needed b The probability that Emerson loses three checker games in a row is Round to four decimal places as needed c The probability that Emerson loses three checker games in a row but does not lose four in a row is Round to fo

A manufacturer wishes to estimate the proportion of washing machines leaving the factory that is defective How large a sample should she check in order to be 94 percent confident that the true proportion is estimated to within 0 015 O 3928 01 O 32 O 3927

Use the given degree of confidence and sample data to construct a confidence interval for the population mean A laboratory tested 90 chicken eggs and found that the mean amount of cholesterol was 225 milligrams with 13 4 milligrams Construct a 95 percent confidence interval for the true mean cholesterol content of all such eggs O 222 228 O 223 229 O 221 227 O 221 228

Use the given degree of confidence and sample data to construct a confidence interval for the population proportion p Of 100 adults selected randomly from one town 63 have health insurance Find a 90 confidence interval for the true proportion of all adults in the town who have health insurance O 0 518 p 0 742 0 535 p 0 725 O 0 551 p 0 709

You select a family with three children If M represents a male child and F represents a female child the set of equally likely outcomes for the children s genders is MMM MMF MFM MFF FMM FMF FFM FFF Find the probability of selecting a family with 4 male children P 4 male children Type an integer or a simplified fraction WIN

The density of people per square kilometer for African countries is in table 2 2 11 Density of people 2013 Create a frequency distribution and a relative frequency distribution using 7 classes 3 42 NNCANSNGUNA 29 79 157 72 69 12 45 630 72 29 341 42 16 2 13 6 415 18 103 9 123 65 8 176 3 3 37 26 a Since data were collected for Select an answer variable s the correct graph to make is a Select an answer

10 pes We bought a pack of 10 lightbulbs from a garage sale and o of them don t work If we were to collect a random sample of 2 lightbulbs no replacement and count the number of lightbulbs that are defective create a probability distribution for this discrete random variable a First define your random variable X The random variable X will be defined as the number of defective lightbulbs in a random sample of 2 lightbulbs chosen without replacement from the pack of 10 bulbs X can take from values 0 to 2 where X 0 means both bulbs are working X 1 means one bulbs is working and X 2 means both bulbs are defective b Next showing work for all of your probabilities in the space below fill in the table below round probabilities to 4 decimal places X P X total of bulbs 10 of defective bulbs 6 of working bulbs 4 probability of 2 bulbs working 4c 100 c What do the probabilities sum up to 70 1333 bc 4c 0 53332 probability of 1 bulbs working IPC 6c probability of 2 bulbs working 10 0 3333 1 0 1333 0 5333 0 3333

5 In an Oreo factory the mean mass of a cookie is given as 40 g For quality control the standard deviation is 2 g a If 10 000 cookies were produced how many cookies are within 5 g of the mean b Cookies are rejected if they weigh more than 44 g or less than 36 g How many cookies would you expect to be rejected in a sample of 10 000 cookies

In Australia in 1995 of the 2907 indigenous people in prison 17 of them died In that same year of the 14501 non indigenous people in prison 42 of them died Aboriginal deaths in 2013 a Find the probability that an indigenous person in prison dies Enter your answer as a fraction Enter your answer rounded to 3 decimal places b Find the probability that a non indigenous person in prison dies Enter your answer as a fraction Enter your answer rounded to 3 decimal places

12 Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y f x y Jxe x20 and y 0 0 otherwise a What is the probability that the lifetime X of the first component exceeds 3 AUX b What are the marginal pdf s of X and Y Are the two lifetimes independent Explain c What is the probability that the lifetime of at least one component exceeds 3

27 Annie and Alvie have agreed to meet for lunch between noon 0 00 P M and 1 00 PM Denote Annie s arrival time by X Alvie s by Y and suppose X and Y are inde pendent with pdf s 1 x Sy y 3r 0 x 1 0 otherwise 2y 0sys1 0 therwise

Solve for v 4 v13 8 V MEN

A person in a casino decides to play 3 games of blackjack Let W denote a win and L denote a loss Define the event A as the person wins at least one game of blackjack What are the possible outcomes for this event O W LW LLW Ow ww www O www WWL WLW WLL LWW LWL LLW LLL O www WWL WLW WLL LWW LWL LLW O WWL LWL LLW

5 Let X denote the number of Canon SLR cameras sold during a particular week by a certain store The pmf of X is 0 1 1 2 2 Px x 25 15 Sixty percent of all customers who purchase these cameras also buy an extended warranty Let Y denote the number of purchasers during this week who buy an extended warranty 3 3 4 a What is P X 4 Y 2 Hint This probability equals P Y 2 X 4 P X 4 now think of the four purchases as four trials of a binomial experiment with success on a trial corresponding to buying an extended warranty Calculate P X Y b c Determine the joint pmf of X and Y and then the marginal pmf of Y

2 A large but sparsely populated county has two small hospitals one at the south end of the county and the other at the north end The south hospital s emergency room has four beds whereas the north hospital s emer gency room has only three beds Let X denote the num ber of south beds occupied at a particular time on a given day and let y denote the number of north beds occupied at the same time on the same day Suppose that these two rv s are independent that the pmf of X puts probability masses 1 2 3 2 and 2 on the x values 0 1 2 3 and 4 respectively and that the pmf of Y distributes probabilities 1 3 4 and 2 on the y values 0 1 2 and 3 respectively a Display the joint pmf of X and Y in a joint probabil ity table b Compute P X 1 and Y 1 by adding probabilities from the joint pmf and verify that this equals the product of P XS 1 and P Y 1 c Express the event that the total number of beds occu pied at the two hospitals combined is at most 1 in terms of X and Y and then calculate this probability d What is the probability that at least one of the two hospitals has no beds occupied

point s possible The data on the right represent the number of live multiple delivery births three or more babies in a particular year for women 15 to 54 years old Use the data to complete parts a through d below Age 15 19 20 24 25 29 30 34 35 39 40 44 45 54 Number of Multiple Births Find the probability that a randomly selected multiple birth for women 15 54 years old involved a mother who was at least 40 years old P at least 40 Type an integer or decimal rounded to three decimal places as needed 92 505 1630 2832 1847 379 117 Interpret this result Select the correct choice below and fill in the answer box to complete your choice Type a whole number OA If 1000 multiple births for women 15 54 years old were randomly selected exactly of them would involve a mother who was at least 40 years old OB If 1000 multiple births for women 15 54 years old were randomly selected we would expect about of them to involve a mother who was at least 40 years old an s a multiple birth involving a mother who was at least 40 years old unusual A No because the probability of a multiple birth involving a mother who was at least 40 years old is greater than 0 05 OB Yes because the probability of a multiple birth involving a mother who was a least 40 years old is greater than 0 05 OC No because the probability of a multiple birth involving a mother who was at least 40 years old is less than 0 05 Yes because the probability of a multiple birth involving a mother who was a

Determine whether the events E and F are independent or dependent Justify your answer a E A person living at least 70 years F The same person regularly handling venomous snakes OA E and F are dependent because living at least 70 years has no effect on the probability of a person regularly handling venomous snakes OB E and F are independent because regularly handling venomous snakes has no effect on the probability of a person living at least 70 years OC E and F are dependent because regularly handling venomous snakes can affect the probability of a person living at least 70 years OD E and F are independent because living at least 70 years has no effect on the probability of a person regularly handling venomous snakes b E A randomly selected person coloring her hair black F A different randomly selected person coloring her hair blond OA E can affect the probability of F even if the two people are randomly selected so the events are dependent OB E can affect the probability of F because the people were randomly selected so the events are dependent OC E cannot affect F because person 1 coloring her hair black could never occur so the events are neither dependent nor independent OD E cannot affect F and vice versa because the people were randomly selected so the events are independent