Review Problems: Week 5
1. [OIS 2.36] Is it worth it? Andy is always looking for ways to make money fast. Lately, he has been trying to make money by gambling. Here is the game he is considering playing: The game costs $2 to play. He draws a card from a deck. If he gets a number card (2-10), he wins nothing. For any face card (jack, queen or king), he wins $3. For any ace, he wins $5, and he wins an extra $20 if he draws the ace of clubs.
a) Create a probability model for Andy's expected profit per game.
b) Compute the expected value of Andy's profit from playing a single game.
c) Compute the standard deviation of his expected profit.
d) Would you recommend this game to Andy as a good way to make money? Explain.
e) Now suppose Andy plays this game twice. Construct a probability model for his expected profit from playing the game twice.
f) Compute the expected value and standard deviation for his profit when playing the game twice.
g) For part f, did you need to use the probability model from part e? Explain.
2. [OIS 2.38] Roulette The game of roulette involves spinning a wheel with 38 slots: 18 red, 18 black, and 2 green. A ball is spun onto the wheel and will eventually land in a slot, where each slot has an equal chance of capturing the ball. Gamblers can place bets on red or black. If the ball lands on their color, they double their money. If it lands on another color, they lose their money.
a) Construct a probability model for your winnings when betting $1 on red.
b) Compute the expected value and standard deviation of your winnings when betting $1 on red.
c) Compute the expected value and standard deviation of your profit when betting $1 on red.
d) To answer part c, did you need to create a new probability model? Explain.
e) Compute the expected value and standard deviation of your total winnings when betting $3 on red in a single round.
f) Compute the expected value and standard deviation of your total winnings when you bet $1 in three consecutive rounds.
g) If you had $100 to play gamble at the roulette table, would you be better off betting $100 on a single round or making 100 $1 bets on 100 different rounds? Explain.
h) Should you play this game at all? Explain.
3. [OIS 3.23] Married women The 2010 American Community Survey conducted by the U.S. Census Bureau estimates that 47.1% of women ages 15 years and over are married.
a) If you randomly select three women above the age of 14, compute the probability that:
i) the third woman selected is the only one who is married.
ii) all three are married.
iii) none are married.
iv) at least one is married.
v) exactly one is married.
vi) at most one is married.
b) On average, when you randomly select three women, who many would you expect to be married?
c) On average, how many women would you expect to sample before selecting a married woman?
4. [OIS 3.27] Underage drinking The Substance Abuse and Mental Health Services Administration estimated that 69.7% of 18–20-year-olds consumed alcoholic beverages during 2008.
a) If you randomly sample ten 18–20-year-olds from the U.S. population, is it appropriate to use a binomial model to compute the probability that exactly six of the 10 consumed alcoholic beverages during 2008? Explain.
b) If you randomly sample ten 18–20-year-olds from the U.S. population, compute the probability that:
i) exactly 6 out of 10 randomly sampled 18–20-year-olds had consumed alcoholic beverages during 2008.
ii) at most 6 out of 10 had consumed alcoholic beverages.
iii) at least 6 out of 10 consumed alcoholic beverages.
iv) exactly 4 out of 10 had not consumed an alcoholic beverage.
v) the first person who had not consumed alcoholic beverages is the fifth person you select.
c) Compute the expected value of the number of 18–20-year-olds (out of 10) who had consumed alcoholic beverages.
d) Compute the expected number of 18–20-year-olds you would need to interview before finding one who had not consumed alcoholic beverages.
5. [OIS 3.28] Chickenpox The National Vaccine Information Center estimates that 90% of Americans have had chickenpox by the time they reach adulthood
a) If you take a random sample of 100 American adults, is it appropriate to use a binomial distribution to compute the probability that exactly 97 had chickenpox before they reached adulthood? Explain.
b) If you randomly sample 100 adults from the U.S. population, compute the probability that
i) exactly 97 had chickenpox during childhood.
ii) exactly 3 have not had chickenpox during childhood?
iii) at most 3 have not had chickenpox during childhood.
iv) at least three have not had chickenpox during childhood.
v) the first person you find who did not have chickenpox during childhood is the tenth person you select?
c) Compute the expected value of the number of people you would need to contact until you find someone who had not had chickenpox during childhood.
d) Compute the expected value of the number of people out of a random sample of 100 who did have chickenpox during childhood.
6. Quiz scores A physics instructor gives his students a 10-point multiple-choice quiz, but due to a mix-up the quiz contains questions about history, not physics. Since none of the physics students have studied history, each of them selects answers by randomly guessing. There are 10 questions on the quiz; each question has five possible responses. For example, the first question might be:
Who was the first president of the United States?
George Washington
George W. Bush
George Foreman
George Clooney
Boy George
a) Compute the probability that a student gets no questions correct.
b) Compute the probability that a student gets exactly three questions correct.
c) Compute the probability that a student gets no more than three questions correct.
d) Compute the probability that a student gets at least three questions correct.
e) On average, how many questions do you expect a student will get correct on this quiz?
f) With what standard deviation?
g) What probability model did you use to answer the previous six questions? Explain why that model was appropriate.
7. Spam According to Google (http://bit.ly/2z5ZMy), 90% of all e-mail messages sent across the Internet are spam. A system administrator randomly selects e-mail messages from among the millions sent on a large network.
a) Compute the probability that the first legitimate e-mail she finds is the seventh message selected.
b) Compute the probability that the first legitimate e-mail is the seventh or eighth message selected.
c) Compute the probability that she finds a legitimate e-mail among the first eight messages.
d) On average, how many messages will she need to select before finding a legitimate message?
e) What probability model did you use to answer the previous three questions? Explain why this model was appropriate.
8. DVRs According to Nielsen Research Corporation, as of February 2011, 39.7% of all households in the United States had a DVR (digital video recorder). You are hired to recruit a small group of DVR users to participate in a focus group. You randomly dial phone numbers until you reach a sufficient number of households with a DVR.
a) Compute the probability that:
i) the first household with a DVR is at the fifth number you call.
ii) among the first five households, all of them have DVR.
iii) among the first five households, exactly two of them have a DVR.
iv) among the first five households, at most two of them have a DVR.
b) In a group of 100 randomly selected households:
i) how many (on average) do you expect to have a DVR?
ii) With what standard deviation?
c) On average, how many households do you expect to call before finding one with a DVR?
9. Tim Eyman According to the Washington Secretary of State (vote.wa.gov), in the election held on November 3, 2009, exactly 44.71% of Snohomish County voters voted in favor of Initiative 1033, a ballot measure supported by Mukilteo resident Tim Eyman. If Tim Eyman wishes to contact a few of the voters who voted in favor of this initiative to ask them to support a similar ballot measure this year, he might hire a consultant to randomly select Snohomish county voters from among those who participated in the November 3, 2009, election. (Whether or not someone voted is a public record, but how they voted is not, so you need to contact them to ask whether or not they voted in favor of the measure.)
a) Compute the probability that:
i) the first I-1033 supporter found is the fourth person contacted.
ii) among five Snohomish County voters, all of them voted in favor of I-1033.
iii) among 12 Snohomish County voters, exactly five of them voted in favor of I-1033.
iv) among 12 Snohomish County voters, at most five of them voted in favor of I-1033.
b) In a group of 12 randomly selected Snohomish County voters:
i) how many (on average) do you expect to have would have voted in favor of I-1033?
ii) With what standard deviation?
c) On average, how many Snohomish County voters do you expect would need to be to contacted before finding one who voted in favor of I-1033?
10. Gambling A friend asks you to play a game involving a (fair) six-sided die. If you roll an even number, you pay him $20. If you roll an odd number, he pays you $10, $30 or $50 depending on the number shown on the die (in other words, 10 times the value shown on the die).
a) Construct a probability model for playing this game once.
b) Compute the expected value and standard deviation for playing this game once.
c) Should you play this game? Explain.
d) To what amount should you change the value you pay your friend when you roll an even number in order to make the game fair?
e) Compute the expected value and standard deviation for playing the game once with this new amount in place.
f) Compute the expected value and standard deviation for playing the new version of the game 10 times.
11. Coins A "fair" coin is one that lands heads up 50% of the time when flipped and tails up 50% of the time when flipped.
a) If you flip a fair coin 10 times, what is the probability that it lands heads up exactly 50% of the time?
b) If you flip a fair coin 10 times, what is the probability that it lands heads up less than 50% of the time?
c) If you flipped a coin 10 times and it landed heads up 4 times, would that lead you to believe the coin is not fair? Explain.
d) If you flipped a coin 10 times and it landed heads up 0 times, would that lead you to believe the coin is not fair? Explain.
e) If you flip a fair coin 10 times, what is the probability that it will land heads up no more than:
i) 3 times?
ii) 2 times?
iii) 1 time?
f) If you flip a coin 10 times, would there be reason to believe it was not fair if it only landed heads up 3 times? Explain.
g) If you flip a coin 10 times, would there be reason to believe it was not fair if it only landed heads up 2 times? Explain.
h) If you flip a coin 10 times, would there be reason to believe it was not fair if it only landed heads up 1 time? Explain.
12. IRS audit According to a January 2012 article by the Associated Press, during 2011 the IRS audited about 1% of all taxpayers earning less than $200,000 a year. (These audits were generally for returns filed in 2011 for income earned during 2010.)
a) On average, how many taxpayers earning less than $200,000 would you need to call at random in order to find one who had been audited by the IRS during 2011?
b) If you randomly select 500 taxpayers who earned less than $200,000, about how many, on average would you expect to have been audited by the IRS during 2011?
c) If you randomly select 500 taxpayers who earned less than $200,000, compute the probability that none of them were audited by the IRS during 2011.
d) If you randomly select 500 taxpayers who earned less than $200,000, compute the probability that at least 10 of them were audited by the IRS during 2011.
e) Assuming the percentage of taxpayers earning less than $200,000 has been about constant over the pat decade (this assumption may not be reasonable), compute the probability that a randomly select taxpayer earning less than $200,000 was:
i) not audited during the previous decade.
ii) audited at least once during the previous decade.
iii) audited more than one during the previous decade.
f) What additional assumption(s) did you need to make in order to answer part e? Were those assumptions reasonable? Explain.
13. Another audit According to a January 2012 article by the Associated Press, during 2011 the IRS audited about 12.5% of all taxpayers earning more than $1,000,000.
a) On average, how many taxpayers earning more than $1,000,000 would you need to call at random in order to find one who had been audited by the IRS during 2011?
b) If you randomly select 50 taxpayers who earned more than $1,000,000, about how many, on average would you expect to have been audited by the IRS during 2011?
c) If you randomly select 50 taxpayers who earned more than $1,000,000, compute the probability that none of them were audited by the IRS during 2011.
d) If you randomly select 50 taxpayers who earned more than $1,000,000, compute the probability that at least 10 of them were audited by the IRS during 2011.
e) According to a March 2012 article from Bloomberg, "In 2009, the IRS created a special unit to examine the tax returns of high-wealth individuals." The article further reported that: "For U.S. taxpayers with adjusted gross incomes between $5 million and $10 million, the audit rate rose to 20.75% [in 2011] from 11.55% [in 2010]. If you randomly select one U.S. taxpayer with adjusted gross income between $5,000,000 and $10,000,000, compute the probability that this taxpayer was audited during both years (2010 and 2011).
f) What assumption(s) did you need to make in order to answer part e? Were those assumptions reasonable? Explain.
g) If you randomly select 10 U.S. taxpayers with adjusted gross incomes between $5,000,000 and $10,000,000, compute the probability that more than one of them was audited:
i) during 2010.
ii) during 2011.
14. Daily Game The Washington State Lottery's Daily Game offers multiple ways to ways to lose money play. The usual way to play involves selecting three digits, each from 0 through 9, and matching those digits (in the same order) with the winning numbers drawn each evening. Another method involves matching those three digits in any order. (For example, if you played 873, then would win if the winning numbers were 873, 837, 783, etc.) If you pay $1 to purchase a ticket, the prize is $80.
a) Construct a probability model for the profit from playing this game once.
b) Compute the expected value and standard deviation of the profit from playing this game once.
c) Should you play this game? Explain.
d) Compute the expected value and standard deviation of the profit from playing this game once a day for an entire year.
15. Jury selection According to the 2010 U.S. Census, 81.7% of Snohomish County residents classify themselves as "white." For many trials, a court randomly selects a jury pool from among the general population of the county, and then jurors are randomly assigned to trials (at which point attorneys are allowed to exclude jurors for various reasons).
a) Compute the probability that a group of 12 jurors selected at random for a trial in Snohomish County:
i) includes no persons of color.
ii) includes only one person of color.
iii) includes at least one person of color.
b) What assumptions did you need to make in order to answer part a? Are those assumptions reasonable? Explain.
c) Juries for misdemeanors (crimes with maximum sentences of less than 12 months) often use 6-person juries. Compute the probability that a group of 6 jurors selected at random for a trial in Snohomish County:
i) includes no persons of color.
ii) includes only one person of color.
iii) includes at least one person of color.
d) On average, how many people of color would you expect to find in a randomly selected 6-person jury pool in Snohomish County?
16. Blood types According to the Red Cross, the prevalence of the four blood types among the general population varies according to ethnic background. Among Caucasians, 45% have Type O blood, 40% have Type A blood, 11% have Type B blood, and the rest have Type AB blood.
a) If you randomly select one Caucasian individual from the general public, compute the probability that his or her blood type:
i) is Type AB.
ii) is Type A or Type B.
iii) is not Type O.
b) The rarest blood type among all ethnic groups is Type AB. On average, how many Caucasians would you need to randomly select until you found one person with Type AB blood?
c) If you randomly select 5 Caucasians from the general public, compute the probability that:
i) all of them have Type O blood.
ii) none of them have Type O blood.
iii) exactly one of them has Type O blood.
iv) at most one of them has Type O blood.
v) at least one of them has Type O blood.
d) If 5 Caucasian siblings walk into a clinic and all volunteer to donate blood, compute (if possible) the probability that exactly three of them have Type A blood. (If not possible, explain.)
17. Blood types (continued) According to the Red Cross, the prevalence of the four blood types among the general population varies according to ethnic background. Among Asians, 40% have Type O blood, 27.5% have Type A blood, 25.4% have Type B blood, and rest have Type AB blood.
a) If you randomly select one Asian individual from the general public, compute the probability that his or her blood type:
i) is Type AB.
ii) is Type A or Type B.
iii) is not Type O.
b) The rarest blood type among all ethnic groups is Type AB. On average, how many Asians would you need to randomly select until you found one person with Type AB blood?
c) If you randomly select 10 Asians from the general public, compute the probability that:
i) all of them have Type B blood.
ii) none of them have Type B blood.
iii) exactly one of them has Type B blood.
iv) at most one of them has Type B blood.
v) at least one of them has Type B blood.
d) If 5 Asian siblings walk into a clinic and all volunteer to donate blood, compute (if possible) the probability that at most three of them have Type B blood. (If not possible, explain.)
18. Nerf guns For a class project, a statistics student tested his theory about the regulators found on Nerf guns: that they slow the muzzle velocity of the darts. He collected three Nerf guns that had 22 barrels among them, each barrel individually regulated. He fired a Nerf dart once using each barrel and measured how many inches it traveled, using the same dart on all tests. He then removed the regulators and fired one shot with each of the barrels again. The data he recorded appears below:
barrel regulator no regulator
1 231 252
2 208 245
3 202 251
4 212 265
5 193 210
6 201 234
7 125 155
8 168 141
9 38 74
10 154 231
11 122 103
12 77 123
13 243 262
14 215 252
15 239 221
16 234 268
17 232 245
18 237 252
19 230 254
20 245 259
21 218 249
22 246 262
a) Is this an observational study or an experiment?
b) If there is no difference between firing with and without a regulator, what percentage of all barrels would expect to fire a Nerf dart further without a regulator than with a regulator?
c) How many barrels actually fired further without a regulator?
d) Assuming that there really is no difference between using a regulator and not using a regulator, compute the probability of observing a result at least as extreme as the one observed here?
e) Are the results of this study unlikely to be observed by chance?
f) What can you conclude about the student's original question?
19. Seatbelts According to the Washington Traffic Safety Commission, 97.5% of drivers in Washington state wear seatbelts, the highest rate in the nation.
a) On average, how many Washington drivers would you need to observe until you found one not wearing a seatbelt?
b) Washington is one of 12 states that outlaws sobriety checkpoints. If these checkpoints were legal in Washington, compute the probability that:
i) the first driver at the checkpoint not wearing a seatbelt is the tenth driver stopped.
ii) among the first 10 drivers, all of them are wearing seatbelts.
iii) among the first 10 drivers, at least one is not wearing a seatbelt.
iv) among the first 10 drivers, at least two are not wearing a seatbelt.
c) What did you need to assume in order to answer part b? Were the assumption(s) reasonable? Explain.
20. Gubernatorial election In the 2012 Washington state primary election, 46.3% of the voters who participated cast their vote for Jay Inslee, while 43.3% cast their vote for Rob McKenna. The
remainder voted for another candidate. We want to randomly select voters from among those who participated in the primary election to interview them about how they plan to vote in the general election.
a) Compute the probability that the first person selected voted for a candidate other than Inslee or McKenna.
b) Compute the probability that the fourth person selected is the first person to have voted for a candidate other than Inslee or McKenna.
c) Compute the probability that you need to select at most four people before you find someone who voted for a candidate other than Inslee or McKenna.
d) Compute the probability that you need to select at least four people before you find someone who voted for a candidate other than Inslee or McKenna.
e) On average, how many voters would you need to select randomly before finding one who voted for a candidate other than Inslee or McKenna?
f) Which probability model did you use to answer the preceding questions?
g) What condition(s) do you need to check before using that model?
21. Gubernatorial again Refer to the information provided in the previous problem.
a) Compute the probability that among 15 randomly selected voters, exactly 8 voted for McKenna.
b) Compute the probability that among 15 randomly selected voters, 8 or 9 voted for McKenna.
c) Compute the probability that among 15 randomly selected voters, at most 8 voted for McKenna.
d) Compute the probability that among 15 randomly selected voters, at least 8 voted for McKenna.
e) Compute the probability that among 15 randomly selected voters, at least one voted for a candidate other than Inslee or McKenna.
f) Among 15 randomly selected voters, on average, how many voters would you expect to have voted for McKenna?
g) Which probability model did you use to answer the preceding questions?
h) What condition(s) do you need to check before using that model?
22. Legalizing marijuana On November 6, 2012, Washington residents approved an initiative that legalized the sale of marijuana. In Colorado, 53.3% of voters in that state voted in favor of Amendment 64, which legalizes recreational marijuana use in Colorado. If you wanted to interview Colorado voters who voted in favor of the amendment, you could randomly select Colorado voters from among those who took part in the 2012 election. (Whether or not someone voted is public record, but how they voted is not, so we need to contact them to ask if they voted in favor of the measure.)
a) Compute the probability that:
i. the first Amendment 64 supporter you find is the sixth person you contact.
ii. among seven Colorado voters, none of them voted in favor of Amendment 64.
iii. among seven Colorado voters, exactly three of them voted in favor of Amendment 64.
iv. among seven Colorado voters, at most three of them voted in favor of Amendment 64.
v. among seven Colorado voters, at least three of them voted in favor of Amendment 64.
b) In a group of 250 randomly selected Colorado voters:
i. how many (on average) do you expect to have voted in favor of Amendment 64?
ii. With what standard deviation?
c) On average, how many Colorado voters would you need to contact until you find one that voted against Amendment 64?
23. Liquor sales On Tuesday, November 8, 2011, Washington residents voted on an initiative (I-1183) that abolished state liquor stores and allowed large retailers such as Costco to sell alcoholic beverages other than beer and wine. the preivous year, two similar initiatives were on the ballot in Washington; according to the Washington Secretary of State (vote.wa.gov), in the election held on November 2, 2010, 35% of Washington voters voted in favor of Initiative 1100, a different ballot measure concerning the sale of alcoholic beverages. We want to randomly select voters from the 2010 election to find if those who voted in favor of the 2010 initiative also voted in favor of the 2011 initiative. (Whether or not someone voted is public record, but how they voted is not, so you need to contact them to ask if they voted in favor of the measure.)
a) Compute the probability that:
i. the first I-1100 supporter you find is the fourth person you contact.
ii. among six 2010 Washington voters, none of them voted in favor of I-1100.
iii. among six 2010 Washington voters, exactly two of them voted in favor of I-1100.
iv. among six 2010 Washington voters, at most two of them voted in favor of I-1100.
v. among six 2010 Washington voters, at least two of them voted in favor of I-1100.
vi. among 13 2010 Washington voters, a majority of them voted against I-1100.
b) In a group of 35 randomly selected Washington voters:
i. how many (on average) do you expect to have voted in favor of I-1100?
ii. With what standard deviation?
iii. Which probability model did you use to answer the previous two questions?
iv. List the conditions you needed to check before using that probability model and explain why each was (or was not) satisfied in this situation.
c) On average, how many Washington voters do you expect to contact before finding one who voted in favor of I-1100?
24. Oregon Pick 4 The Oregon Lottery's Pick 4 offers multiple ways to ways to lose money play. One way involves selecting four digits, each from 0 through 9, and matching those digits (in the same order) with the winning numbers drawn each evening. If you pay $1 to purchase a ticket, the prize is $5000.
a) Construct a probability model for the profit from playing this game once.
b) Compute the expected value and standard deviation of the profit from playing this game once.
c) Should you play this game? Explain.
d) Compute the expected value and standard deviation of the profit from playing this game once a day for an entire month.
25. Seatbelts again According to the CDC, only 58% of drivers in North Dakota wear seatbelts, the lowest rate in the nation.
a) On average, how many North Dakota drivers would you need to observe until you found one not wearing a seatbelt?
b) Sobriety checkpoints are legal in North Dakota. Compute the probability that:
i) the first driver at a checkpoint not wearing a seatbelt is the fifth driver stopped.
ii) among the first five drivers, all of them are wearing seatbelts.
iii) among the first five drivers, at least one is not wearing a seatbelt.
iv) among the first five drivers, at least two are not wearing a seatbelt.
c) What did you need to assume in order to answer part b? Were the assumption(s) reasonable? Explain.
26. Jury selection again According to the 2010 U.S. Census, 77.3% of Pierce County residents classify themselves as "white." For many trials, a court randomly selects a jury pool from among the general population of the county, and then jurors are randomly assigned to trials (at which point attorneys are allowed to exclude jurors for various reasons).
a) Compute the probability that a group of 12 jurors selected at random for a trial in Pierce County:
i) includes no persons of color.
ii) includes only one person of color.
iii) includes at least one person of color.
b) What assumptions did you need to make in order to answer part a? Are those assumptions reasonable? Explain.
c) Juries for misdemeanors (crimes with maximum sentences of less than 12 months) often use 6-person juries. Compute the probability that a group of 6 jurors selected at random for a trial in Pierce County:
i) includes no persons of color.
ii) includes only one person of color.
iii) includes at least one person of color.
d) On average, how many people of color would you expect to find in a randomly selected 6-person jury pool in Pierce County?
27. Legalizing marijuana again On November 6, 2012, 55.7 % of Washington voters voted to approve Initiative 502, which legalized the sale of marijuana. If you wanted to interview Washington voters who voted in favor of the amendment (in order to seek their opinions about how the new law is being implemented), you could randomly select Washington voters from among those who took part in the 2012 election. (Whether or not someone voted is public record, but how they voted is not, so we need to contact them to ask if they voted in favor of the measure.)
a) Compute the probability that:
i. the first I-502 supporter you find is the fourth person you contact.
ii. among six Washington voters, none of them voted in favor of I-502.
iii. among six Washington voters, exactly three of them voted in favor of I-502.
iv. among six Washington voters, at most three of them voted in favor of I-502.
v. among six Washington voters, at least three of them voted in favor of I-502.
b) In a group of 125 randomly selected Washington voters:
i. how many (on average) do you expect to have voted in favor of I-502?
ii. With what standard deviation?
c) On average, how many Washington voters would you need to contact until you find one that voted against I-502?
28.