Probability Questions and Answers

The distribution of Master s degrees conferred by a university is listed in the table Major Frequency Mathematics English Engineering Business Education 216 207 72 176 222 What is the probability that a randomly selected student graduating with a Master s degree has a major of Engineering Round your answer to three decimal places OA 0 081 B 0 014 OC 0 919 O D 0 986

1 Here are the quiz scores from a period of geometry students at North High School 56 89 98 90 65 32 34 74 95 88 96 57 84 74 48 55 55 74 88 87 91 73 67 76 Find the percentile for someone who scores an 84 What does this value mean

Find the mean of the binomial distribution for which n 50 and p 0 2 needed OA 10 00 B 2 83 C 50 00 OD 25 00 und

A survey of 280 homeless persons showed that 63 were veterans Construct a 90 confidence interval for the proportion of homeless persons who are veterans OA 0 184 0 266 OB 0 167 0 283 OC 0 176 0 274 OD 0 161 0 289

information service you want to test that claim To do so you select a random sample of 45 cell phones of the manufacturer s latest model and record their battery lifetimes Assume it is known that the population standard deviation of the battery lifetimes for that cell phone model is 2 73 hours Based on your sample follow the steps below to construct a 99 confidence interval for the population mean of the battery lifetimes for all phones of the manufacturer s latest model Then state whether the confidence interval you construct contradicts the manufacturer s claim If necessary consult a list of formulas a Click on Take Sample to see the results from your random sample of 45 phones of the manufacturer s latest model Take Sample Sample size 0 Point estimate 0 Population standard deviation 0 Critical value 0 Enter the values of the sample size the point estimate for the population mean the population standard deviation and the critical value you need for your 99 confidence interval Choose the correct critical value from the table of critical values provided When you are done select Compute Compute 0 00 Number of phones 45 0 00 Sample mean 2 00 7 38 Sample standard deviation 2 61 4 00 Standard error Margin of error 99 confidence interval b Based on your sample graph the 99 confidence interval for the population mean of the battery lifetimes for all phones of the manufacturer s latest model 99 confidence interval Enter the lower and upper limits on the graph to show your confidence interval For the point enter the manufacturer s claim of 5 99 hours 5 00 6 00 below O No the confidence interval does not contradict the claim The manufacturer s claim of 5 99 hours is inside the 99 confidence interval O No the confidence interval does not contradict the claim The manufacturer s claim of 5 99 hours is outside the 99 confidence interval O Yes the confidence interval contradicts the claim The manufacturer s claim of 5 99 hours is inside the 99 confidence interval Population standard deviation 2 73 8 00 Yes the confidence interval contradicts the claim The manufacturer s claim of 5 99 hours is outside the 99 confidence interval Critical values 0 005 2 576 X 0 010 2 326 20 025 1 960 0 050 1 645 0 100 1 282 X X c Does the 99 confidence interval you constructed contradict the manufacturer s claim Choose the best answer from the choices 10 00 5 10 00 80 M

67 of U S adults have very little confidence in newspapers You randomly select 10 U S adults Find the probability that the number of U S adults who have very little confidence in newspapers is a exactly five b at least six and c less than four a P 5 b P x 6 c P x 4 Round to three decimal places as needed Round to three decimal places as needed Round to three decimal places as needed

K The accompanying table shows asked if they contribute to a retirement savings plan at work Complete parts a and b below Click the icon to view the survey results a Find the probability that a randomly selected worker contributes to a retirement savings plan at work given that the worker is male The probability that a randomly selected worker contributes to a retirement savings plan at work given that the worker is male is Round to three decimal places as needed b Find the probability that a randomly selected worker is female given that the worker contributes to a retirement savings plan at work www The probability that a randomly selected worker is female given that the worker contributes to a retirement savings plan at work is Round to three decimal places as needed Survey Results Male Female Total Contribute 121 146 267 Do not contribute Total 250 250 500 129 104 233 X

Classify the two given samples as independent or depende Sample 1 The weights in pounds of 14 newborn females Sample 2 The weights in pounds of 14 newborn males O dependent

The responses of 1622 voters to a survey about the way the media conducted themselves in a recent politica campaign are shown in the accompanying Pareto chart Find the probability of each event listed in parts a through d below Click the icon to view the Pareto chart a Randomly selecting a person from the sample who did not give the media an A or a B The probability is 0 775 Type an integer or a decimal Round to three decimal places as needed b Randomly selecting a person from the sample who gave the media a grade better than a D The probability is Type an integer or a decimal Round to three decimal places as needed

A card is selected at random from a standard deck of 52 playing cards Find the probability of each event a Randomly selecting a spade or a 2 b Randomly selecting a red suit or a king c Randomly selecting a 4 or a face card a The probability of randomly selecting a spade or a 2 is Type an integer or decimal rounded to three decimal places as needed

You have two die calculate the probability of rolling a sum of two or two even numbers First set up the sample space see previous lessons as a table Second find the probability of rolling a two Third find the probability of rolling two even numbers Fourth determine this event is Fifth use the formula Sixth the probability is 4

Given this table about Tuberculosis TB which is a disease that affects the lungs Most Americans are tested for TB when the are young and when they will be working with youth What would the expression P have TB Test negative represent People Tested for TB Have TB Does not have TB Total Test Positive 198 98 Test Negative 2 9702 total 9800 Select one O a Conditional Relative Frequency Ob None of these O c Joint Relative Frequency O d Marginal Relative Frequency 9704

In a baseball tournament consisting of twelve teams in which each team plays every other team once how many games will there be Select one O a 24 O b 720 O C 15 O d 66

Use the probability distribution to complete parts a through d below The probability distribution of number of televisions per household in a small town X 0 1 3 P x 0 05 0 17 0 46 2 0 32 a Find the probability of randomly selecting a household that has one or two televisions The probability is 0 Type an integer or a decimal Do not round

ng the data given below determine whether it would unusual for a household to have no HD televisions The number of televisions HD per household in a small town Televisions 0 1 2 3 Households 26 434 1404 0 010 0 167 0 540 P x Choose the correct answer below 736 0 283 OA It would be unusual because the probability of having no HD televisions is less than 0 05 B It would not be unusual because the probability of having no HD televisions is more than 0 05 OC It would be unusual because 26 people have no HD televisions in the town OD It would not be unusual because 26 people have no HD televisions in the town

Now fill in the missing probabilities Start F Check Clear Help del Laivo unl E E Now we can use the tree above answer the original question If Hugo or any other player for that matter tests positive what is the probability that he uses steroi

A B AB O Total Rh Negative 45 73 22 49 189 Rh Positive 81 56 8 28 173 Total 126 129 30 77 362 Find the probability of choosing a donor with type A or Rh Negative blood Round to four decimals as needed Provide your answer below

Part 1 of 3 Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of 1 1 kg and a standard deviation of o 4 5 kg Complete parts a through c below a If 1 male college student is randomly selected find the probability that he gains between 0 kg and 3 kg during freshman year The probability is Round to four decimal places as needed

Part 1 of 4 In a science fair project Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field She flipped a coin to select either her right hand or her left hand and then she asked the therapists to identify the selected hand by placing their hand just under Emily s hand without seeing it and without touching it Among 284 trials the touch therapists were correct 136 times Complete parts a through d COLO a Given that Emily used a coin toss to select either her right hand or her left hand what proportion of correct responses would be expected if the touch therapists made random guesses Type an integer or a decimal Do not round

CQ1 Construct a two way table with 100 women and 80 men in which both groups show equal percentages of right handedness

5 51 Independent Suppose a person is chosen at random Use your understanding about the world of basketball to decide whether the event that the person is taller than six feet and the event that the person plays professional basketball are independent or associated Explain

5 17 Birthdays What is the probability that a baby will be born on a Friday OR a Saturday OR a Sunday if all the days of the week are equally likely as birthdays

5 48 College Poll A person is selected randomly from the entire group whose responses are summarized in the table for Exercise 5 47 We want to find the probability that the person selected is a male who said yes Which of the following statements best describes the problem i P Yes Male ii P Malel Yes iii P Male AND Yes

5 8 Random Assignment of Professors A study randomly assigned students attending the Air Force Academy to different pro fessors for Calculus I with equal numbers of students assigned to each professor Some professors were experienced and some were relatively inexperienced Suppose the names of the professors are Peters Parker Diaz Nguyen and Black Suppose Diaz and Black are inexperienced and the others are experienced The researchers reported that the students who had the experienced teachers for Calculus I did better in Calculus II Source Scott E Carrell and James E West Does Professor Quality Matter Evidence from Random Assignment of Students to Professors 2010 a List the equally likely outcomes that could occur for assignment of one student to a professor b Suppose the event of interest event A is that a teacher is experienced List the outcomes that make up event A c What is the probability that a student will be assigned to an experienced teacher d List the outcomes in the complement of event A Describe this complement in words e What is the probability that a student will be assigned to an inexperienced teacher

5 14 Guessing on Tests Consider a multiple choice test with a total of four possible options for each question a What is the probability of guessing correctly on one question Assume that there are three incorrect options and one correct option b What is the probability that a guess on one question will be incorrect

5 7 Criminal Court Judges Criminal cases are assigned to judges andomly The list of the criminal judges for Memphis Tennessee taken from the Tennessee Court System website www tsc state tn us is given in the table Assume that only Carolyn Blackett and Paula Skahan are women and the rest are men If you were a criminal defense attorney in Memphis you might be interested in whether the udge assigned to your case was a man or a woman James C Beasley Carolyn W Blackett Lee V Coffee Chris Craft John T Fowlkes W Otis Higgs James M Lammey Paula L Skahan W Mark Ward

5 10 Which of the following numbers could not be probabilities and why a 125 b 0 74 c 0 001 d 5 61 e 150 TRY 5 11 Playing Cards Example 2 There are four suits clubs diamonds hearts and spades 4 and the following cards appear in each suit ace 2 3 4 5 6 7 8 9 10 jack queen king The jack queen and king are called face cards because they have a drawing of a face on them Diamonds and hearts are red and clubs and spades are black If you draw 1 card randomly from a standard 52 card playing deck what is the probability that it will be a A heart b A red card c An ace d A face card jack queen or king e A three 5 12 Playing Cards Refer to Exercise 5 11 for information about cards If you draw 1 card randomly from a standard 52 card playing deck what is the probability that it will be a A black card b A diamond c A face card jack queen or king d A nine e A king or queen 5 13 Guessing on Tests a On a true false quiz in which you are guessing what is the probability of guessing correctly on one question b What is the probability that a guess on one true false question will be incorrect 5 14 Guessing on Tests Consider a multiple choice test with a total of four possible options for each question a What is the probability of guessing correctly on one question Assume that there are three incorrect options and one correct option b What is the probability that a guess on one question will be incorrect TRY 5 15 Four Children Example 3 The sample space given here shows all possible sequences for a family with 4 children where B stands for boy and G stands for girl GGGG GGGB GGBB GBBB BBBB GGBG GBGB BGBB GBGG GBBG BBGB BGGG BGGB BBBG BGBG BBGG Assume that all of the 16 outcomes are equally likely Find the probability of having the following numbers of girls out of Hint The probability of having 3 girls and a boy is 4 16 or 25 because the second column shows that there are 4 ways to have 3 girls and I boy 5 16 Three Coins The sample shows the possible sequences for flipping three fair coins or flipping one coin three times where H stands for heads and T stands for tails HHH HHT HTH THH Assume that all of the 8 outcomes are equally likely Find the probability of having exactly the following numbers of heads out of the 3 coins a exactly 0 heads b exactly 1 head c exactly 2 heads d exactly 3 heads e What do the four probabilities add up to and why No HTT THT TTH 5 17 Birthdays What is the probability that a baby will be born on a Friday OR a Saturday OR a Sunday if all the days of the week are equally likely as birthdays 5 18 Playing Cards If one card is selected from a well shuffled deck of 52 cards what is the probability that the card will be a club OR a diamond OR a art What is probab of the complement of this event Refer to Exercise 5 11 for information about cards Unsure Yes TTT 5 19 College Poll A StatCrunch poll asked people if college was worth the financial investment They also asked the respondent s gender The table shows a summary of the responses Source StatCrunch Responses to Is college worth it Owner scsurvey All Female Male 45 56 100 96 577 401 722 553 All 101 196 978 1275 a If a person is chosen randomly from the group what is the probability that the person is male b If a person is chosen randomly from the group what is the probability that the person said Yes 5 20 College Poll Refer to the table given for Exercise 5 19 a If a person is chosen randomly what is the probability that the person is female b If a person is chosen randomly what is the probability that the person said No TRY 5 21 College Poll AND Example 4 Refer to the table given for Exercise 5 19 If a person is chosen randomly from the group what is the probability that the person is female AND said Yes 5 22 College Poll AND Refer to the table given for Exercise 5 19 If a person is chosen randomly from the group what is the probability that the person is male AND said No TRY 5 23 College Poll OR Example 5 Refer to the table given for Exercise 5 19 a If a person is chosen randomly from the group what is the probability

5 16 Three Coins The sample shows the possible sequences for flipping three fair coins or flipping one coin three times where H stands for heads and T stands for tails HHH HHT HTH THH HTT THT TTH TTT Assume that all of the 8 outcomes are equally likely Find the probability of having exactly the following numbers of heads out of the 3 coins a exactly 0 heads b exactly 1 head c exactly 2 heads d exactly 3 heads e What do the four probabilities add

a On a true false quiz in which you are guessing what is the probability of guessing correctly on one question b What is the probability that a guess on one true false question will be incorrect

5 1 Simulation Example 1 If we flip a coin 10 times how often do we get 6 or more heads A first step to answering this question would be to simulate 10 flips Use the random number table in Appendix A to simulate flipping a coin 10 times Let odd digits 1 3 5 7 9 represent heads and let even digits 0 2 4 6 8 represent tails Begin with the first digit in the third row a Write the sequence of 10 random digits b Write the sequence of 10 heads and tails Write H for heads and T for tails c How many heads did you get Did you get 6 or more heads

5 11 Playing Cards Example 2 There are four suits clubs diamonds hearts and spades and the following cards appear in each suit ace 2 3 4 5 6 7 8 9 10 jack queen king The jack queen and king are called face cards because they have a drawing of a face on them Diamonds and hearts are red and clubs and spades are black If you draw 1 card randomly from a standard 52 card playing deck what is the probability that it will be a A heart b A red card c An ace d A face card jack queen or king A throa

5 9 Which of the following numbers could not be probabilitie and why a 5 63 b 0 063 c 0 063 d 163 e 1 63

A normal population has mean 51 and standard deviation o 7 What is the 85th percentile of the population Use the TI 84 Plus calculator Round the answer to at least one decimal place The 85th percentile of the population is X

insomniacs by an average of 2 7 hours with a variance of 0 30 A governr group conducts an independent study on the new pill to see if the varianc of increase in sleeping time is different from the advertised variance The government group tests 23 insomniacs chosen at random with the nev The insomniacs in the sample report a mean daily sleeping increase of hours with a variance of 0 58 Is there enough evidence at the 0 05 leve significance to reject the claim made in the advertisement Assume th the increases in sleeping times provided by the new pill are approxima normally distributed Perform a two tailed test Then complete the parts below Carry your intermediate computations to three or more decimal places necessary consult a list of formulas a State the null hypothesis Ho and the alternative hypothesis H HO H b Determine the type of test statistic to use Choose one c Find the value of the test statistic Round to three or more decimal places d Find the two critical values Round to three or more decimal places and e Can we reject the claim that the variance of increase in sleeping time using the new pill is as advertised OYes No H x X 0 a X S 0 0 OSO 020 P 0 0 00 0

Consider the following Venn diagram Q

Consider the following Venn diagram C C

In a running competition a bronze silver and gold medal must be given to the top three girls and top three boys If 12 boys and 7 girls are competing how many different ways could the six medals possibly be given out

what is the mean of the discrete vendom veriable X O with the following proberbility distribution 13 x 1 3 2 3

Consider the following Venn diagram B Q

8000 the following Venn diagram A Pr A is closest to 05 18 07 14 07 1 017 011 20 04 9 07 16 014 17 000000 Pr B is closest to 08 18 05 7 07 16 01 11 06 14 06 9 13 20 10 17 Pr A U B is closest to 08 14 6 9 18 20 B

In order to spice up your investment portfolio you decided on the morning of March 19 2004 to ignore your broker s cautious advice and purchase three tech stocks and two non tech stocks at random from the following selection prices as of March 19 2004 recommended by your best friend Tech Stocks Change Non Tech Stocks 0 07 PENN Penn Gaming NY Nortel MSFT Microsoft CSCO Cisco NOK Nokia INTC Intel EBAY eBay Open 5 46 24 89 23 17 20 77 27 20 69 34 0 26 0 67 0 75 0 71 1 04 DUK Duke KSE Keyspan ED Con Ed MO Altria Group AACE Ace Cash Express Open 27 09 22 04 37 75 44 51 55 04 27 09 Change 0 71 0 01 0 18 0 39 0 37 0 49 a 1 point How many possible selections are possible b 2 points You tend to have bad luck with stocks they usually go down the moment you buy them How many of the selections in part a consist only of stocks that declined in value by the end of the day What is the probability that all of your stocks declined in value by the end of the day

Harry has a 1 year ARM for 75 000 over a 25 year term The margin is 2 and the index rate starts out at 2 5 and increases to 3 5 at the first adjustment The balance of principal at the end of the first year is 73 338 52 Find the amount of monthly payment for the first month of the first year If necessary use the accompanying table of monthly payments Round to the nearest cent Click the icon to view the monthly payments table OA 489 42 OB 416 87 O C 388 14 D None of the above is correct

Consider the following distribution 1 20 rating n 4821 0 062 0 25 0 01 0 562 0 81 4 15 0 Imagine an experiment where we will randomly draw a single observation from within this collection The probability that the value is between 4 068 and 0 005 is closest to 00 75 00 25 D01 00 5 D0 9 1 10 5 Imagine an experiment where we will randomly draw two observations from within this collection The probability that both values are between 20 093 and 2 359 is closest to

ghts of college football players are normally distributed with a mean of 200 pounds and a standa deviation of 50 pounds If a college football player is randomly selected find the probability that he weig between 170 and 220 pounds Round to four decimal places A 0 0703 OB 0 1554 OC 0 2257 D 0 3812 www

selected find the probability that the person is a man or a heavy smoker Round to three decima bits of a group of asthma sufferers If one of the 1156 people is places as needed Men Women Total A 0 511 B 0 596 OC 0 557 D 0 554 Nonsmoker 431 382 813 Occasional Smoker 50 48 98 Regular Smoker 71 86 157 Heavy Smoker 49 39 88 Total 601 555 1156

ade W Una test of agender selection technique results Considered of 254 baby girls 5 baby boys Based on this result what is the probability of a girl born to a Couple using this technique The probality that a girl will we born using this technique is approximately answer needs to have Ba be rounded 3 decimal placer

Can the FUN argument to apply be user defined O Yes No

Suppose one wants to matrix multiply a matrix A by a matrix B in R Which of the following conditions must be true Oncol A nrow A Oncol A ncol B ncol B nrow B

When created using matrix are matrices filled by row by default O Yes O No

Are matrices in R stored as vectors Yes No