AP Stats Sapling Extra Practice Chapter 5

n a large business hotel, 40% of the guests read the Los Angeles Times. Only 25% read the Wall Street Journal. Five percent of guests read both papers. Suppose we select a hotel guest at random and record which of the two papers the person reads, if either. Using the events L: reads the Los Angeles Times and W: reads the Wall Street Journal, find 𝑃(LC∩W)P(LC∩W).

0.2

In the game of Scrabble, each player begins by randomly selecting 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. We want to perform a simulation to determine the probability that a player will randomly select 7 vowels. 0069405977196646544120903623712272553340 Using the random digits provided, how many vowels are selected in one trial of the simulation?

2

In January 2017, 52% of U.S. senators were Republicans and the rest were Democrats or Independents. Twenty-one percent of the senators were females, and 47% of the senators were male republicans. Suppose we select one of these senators at random. Define events R: is a Republican and M: is male. Find 𝑃(R∪M)P(R∪M) Interpret this value in context.

0.84. The probability of randomly selecting a senator who is Republican or a male is 0.84.

The Pew Research Center asked a random sample of 2024 adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data.Suppose we select one of the survey respondents at random. Age18-3435-5455+TotalType of cell phoneiPhone169171127467Android214189100503Other1342776431054Total5176378702024 What is the probability that the person is age 18 to 34 or owns an iPhone?

169+214+134+171+1272024=0.403

Consider that, in the U.S. criminal justice system, defendants are presumed to be innocent until proven guilty. Imagine that you are the suspect taking the lie detector test. Which is more unjust: an innocent person being convicted or a guilty person going free? Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

A false positive is a more serious error because it could result in someone potentially being convicted of a crime that they did not commit.

Abigail, Bobby, Carlos, DeAnna, and Emily go to the bagel shop for lunch every Thursday. Each time, they randomly pick 2 of the group to pay for lunch by drawing names from a hat. Give a probability model for this chance process.

Sample space = {Abigail/Bobby, Abigail/Carlos, Abigail/DeAnna, Abigail/Emily, Bobby/Carlos, Bobby/DeAnna, Bobby/Emily, Carlos/DeAnna, Carlos/Emily, DeAnna/Emily}. Each of the 10 events has probability 1/10.

Imagine tossing a fair coin 3 times. Give a probability model for this chance process.

Sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Each of the events has probability 1/8.

Canada has two official languages, English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is a distribution of responses, combining many separate languages from the broad Asia/Pacific region. LanguageEnglishFrenchAsian/PacificOtherProbability0.630.220.060.09 Is this a valid probability model? Why or why not?

Yes, because each probability is between 0 and 1 and all the probabilities sum to 1.

Suppose C and D are two events such that 𝑃(C)=0.6P(C)=0.6, 𝑃(D)=0.45P(D)=0.45, and 𝑃(C∩𝐷)=0.3P(C∩D)=0.3. Are events C and D independent? Justify your answer?

No, because (0.6)(0.45)≠0.3

There are many married couples in which the husband and wife both carry a gene for cystic fibrosis but don't have the disease themselves. Suppose we select one of these couples at random. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25. If researchers randomly select 4 such couples, is one of these couples guaranteed to have a first child who develops cystic fibrosis? Why or why not?

No. If researchers randomly select 4 such couples, there will not necessarily be exactly one that has a first child born with cystic fibrosis.

A statistics class with 30 students has 10 males and 20 females. Suppose you choose 3 of the students in the class at random. Find the probability that all three are female.

2030⋅1929⋅18282030⋅1929⋅1828 = 0.281

Suppose that 10% of adults belong to health clubs, and 40% of these health club members go to the club at least twice a week. Find the probability that a randomly selected adult belongs to a health club and goes there at least twice a week.

(0.10)(0.40)=0.04=4(0.10)(0.40)=0.04=4%

A company that offers courses to prepare students for the Graduate Management Admission Test (GMAT) has collected the following information about its customers: 20% are undergraduate students in business, 15% are undergraduate students in other fields of study, and 60% are college graduates who are currently employed. Choose a customer at random. What must be the probability that the customer is a college graduate who is not currently employed?

0.05

Here is the distribution of the adjusted gross income (in thousands of dollars) reported on individual federal income tax returns in a recent year: Income< 2525-4950-99100-499 500Probability0.4310.2480.2150.1000.006 What is the probability that a randomly chosen return shows an adjusted gross income of $50,000 or more?

0.321

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.Suppose we select one of the students at random. Grade4th grade5th grade6th gradeTotalMost importantGrades495069168Athletic24363898Popular19222869Total92108135335 What is the probability that the student is a sixth-grader or rated good grades as important?

69+38+28+49+50335=0.609

A professional tennis player claims to get 90% of her second serves in. In a recent match, the player missed 5 of her first 20 second serves. Is this a surprising result if the player's claim is true? Assume that the player has a 0.10 probability of missing each second serve. We want to carry out a simulation to estimate the probability that she would miss 5 or more of her first 20 second serves. The dotplot displays the number of second serves missed by the player out of the first 20 second serves in 100 simulated matches. Use the results of the simulation to estimate the probability that the player would miss 5 or more of her first 20 second serves in a match.

7/100 = 7%

A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits - clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jacks, queens, and kings are referred to as "face cards." Imagine that we shuffle the deck thoroughly and deal one card. Define events F: getting a face card and H: getting a heart. The two-way table summarizes the sample space for this chance process. CardFace cardNon-face cardTotalSuitHeart31013Non-heart93039Total124052 Find 𝑃(HC and F)

952=0.173952=0.173If the deck is shuffled thoroughly and one card is dealt, the probability that the dealt card is not a heart and is a face card is 0.173.

There are many married couples in which the husband and wife both carry a gene for cystic fibrosis but don't have the disease themselves. Suppose we select one of these couples at random. According to the laws of genetics, the probability that their first child will develop cystic fibrosis is 0.25. Interpret this probability as a long-run relative frequency.

If you take a very large random sample of couples in which the husband and wife both carry a gene for cystic fibrosis, but do not have the disease themselves, about 25% of such couples will find that their first child develops cystic fibrosis.

Sometimes police use a lie detector test to help determine whether a suspect is telling the truth. A lie detector isn't foolproof - sometimes it suggests that a person is lying when he or she is actually telling the truth (a "false positive"). Other times, the test says that the suspect is being truthful when he or she is actually lying (a "false negative"). For one brand of lie detector, the probability of a false positive is 0.08. Explain what this probability means.

If you take a very large random sample of truthful people, about 8% of the time the test will indicate that the truthful person is lying.

The figure shows the results of two different sets of 5000 coin tosses. Explain what this graph tells you about chance behavior in the short run and long run.

In the short run, there was quite a bit of variability in the proportion of heads. In the long run, this proportion became less variable and settled down around 0.50 for both sets of 5000 tosses.

A website claims that 10% of U.S. adults are left-handed. A researcher believes that this figure is too low. She decides to test this claim by taking a random sample of 20 U.S. adults and recording how many are left-handed. Four of the adults are left-handed. Does this result give convincing evidence that the website's 10% claim is too low? To find out, we want to perform a simulation to estimate the probability of getting 4 or more left-handed people in a random sample of size 20 from a very large population in which 10% of the people are left-handed. Let 00 to 09 indicate left-handed and 10 to 99 represent right-handed. Move left to right across a row in Table D. Each pair of digits represents one person. Keep going until you get 20 different pairs of digits. Record how many people in this simulated sample are left-handed. Repeat this process many, many times. Find the proportion of trials in which 4 or more people in the simulated sample were left-handed. Does the problem describe a legitimate simulation design? Justify your answer.

No, the problem does not describe a legitimate simulation design. As written, the simulation calls for "20 different pairs of digits" to be selected from Table D. Because the digits in the simulation represent left-handedness or right-handedness, the digits should be selected with replacement (repeats should be allowed).

Imagine tossing a fair coin 3 times. Define event B as getting more heads than tails. Find P(B).

P(B) = 4/8 = 0.50

How is the hatching of water python eggs influenced by the temperature of a snake's nest? Researchers randomly assigned newly laid eggs to one of three water temperatures: cold, neutral, or hot. Hot duplicates the extra warmth provided by the mother python, and cold duplicates the absence of the mother.Suppose we select one of the eggs at random. Nest temperatureColdNeutralHotHatching statusHatched163875Didn't hatch111829 Given that the chosen egg was assigned to hot water, what is the probability that it hatched?

P(hatched ∣∣ assigned to hot water) = 7575+29=75104=0.721

Canada has two official languages, English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is a distribution of responses, combining many separate languages from the broad Asia/Pacific region. LanguageEnglishFrenchAsian/PacificOtherProbability0.630.220.060.09 What is the probability that the chosen person's mother tongue is not English?

P(not English) = 0.37

How is the hatching of water python eggs influenced by the temperature of a snake's nest? Researchers randomly assigned newly laid eggs to one of three water temperatures: cold, neutral, or hot. Hot duplicates the extra warmth provided by the mother python, and cold duplicates the absence of the mother. Suppose we select one of the eggs at random. Nest temperatureColdNeutralHotHatching statusHatched163875Didn't hatch111829 If the chosen egg hatched, what is the probability that it was not assigned to hot water?

P(not assigned to hot water ∣∣ hatched) = 16+3816+38+75=54129=0.419

A very good professional baseball player gets a hit about 35% of the time over an entire season. After the player failed to hit safely in six straight at-bats, a TV commentator said, "He is due for a hit." Is the commentator's statement right or wrong? Justify your answer.

The commentator is wrong. He is incorrectly applying the law of large numbers to a small number of at-bats for the player.

A professional tennis player claims to get 90% of her second serves in. In a recent match, the player missed 5 of her first 20 second serves. Is this a surprising result if the player's claim is true? Assume that the player has a 0.10 probability of missing each second serve. We want to carry out a simulation to estimate the probability that she would miss 5 or more of her first 20 second serves. The dotplot displays the number of second serves missed by the player out of the first 20 second serves in 100 simulated matches. Which of the following choices is not a correct interpretation of the dot at 6?

The tennis player missed her second serve 6 times out of 20 during one tennis match.

Lactose intolerance causes difficulty in digesting dairy products that contain lactose (milk sugar). It is particularly common among people of African and Asian ancestry. In the United States (not including other groups and people who consider themselves to belong to more than one race), 82% of the population is White, 14% is Black, and 4% is Asian. Moreover, 15% of Whites, 70% of Blacks, and 90% of Asians are lactose intolerant. Suppose we select a U.S. person at random and find that the person is lactose intolerant. What's the probability that he or she is Asian?

𝑃(Asian∣lactose intolerant)=(0.04)(0.90)(0.82)(0.15)+(0.14)(0.70)+(0.04)(0.90)=0.14

A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits - clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jacks, queens, and kings are referred to as "face cards." Imagine that we shuffle the deck thoroughly and deal one card. Define events F: getting a face card and H: getting a heart. The two-way table summarizes the sample space for this chance process. CardFace cardNon-face cardTotalSuitHeart31013Non-heart93039Total124052 Find 𝑃(HC and F)

0.173

Suppose that you roll a fair, six-sided die 10 times. What's the probability that you get at least one 6?

1−(5/6)10=0.8385

The National Household Travel Survey gathers data on the time of day when people begin a trip in their car or other vehicle. Choose a trip at random and record the time at which the trip started. Here is an assignment of probabilities for the outcomes: Time of day10 p.m. - 12:59 a.m.1 a.m. - 5:59 a.m.6 a.m. - 8:59 a.m.Probability0.0400.0330.144Time of Day9 a.m. - 12:59 p.m.1 p.m. - 3:59 p.m.4 p.m. - 6:59 p.m.Probability0.2340.208?Time of day7 p.m. - 9:59 p.m.Probability0.123 What probability should replace "?" in the table?

0.218

Suppose C and D are two events such that 𝑃(C)=0.6P(C)=0.6, 𝑃(D)=0.45P(D)=0.45, and 𝑃(C∪D)P(C∪D) = 0.75. Find 𝑃(C∩D)

0.3

A company that offers courses to prepare students for the Graduate Management Admission Test (GMAT) has collected the following information about its customers: 20% are undergraduate students in business, 15% are undergraduate students in other fields of study, and 60% are college graduates who are currently employed. Choose a customer at random. Find the probability that the customer is currently an undergraduate.

0.35

A recent census at a major university revealed that 60% of its students mainly used Macs. The rest mainly used PCs. At the time of the census, 67% of the school's students were undergraduates. The rest were graduate students. In the census, 23% of respondents were graduate students and used a Mac as their main computer. Suppose we select a student at random from among those who were part of the census and learn that the person mainly uses a Mac. Find the probability that the person is a graduate student.

0.383

In a large business hotel, 40% of the guests read the Los Angeles Times. Only 25% read the Wall Street Journal. Five percent of guests read both papers. Suppose we select a hotel guest at random and record which of the two papers the person reads, if either. What's the probability that the person reads the Los Angeles Times or the Wall Street Journal?

0.6

he National Household Travel Survey gathers data on the time of day when people begin a trip in their car or other vehicle. Choose a trip at random and record the time at which the trip started. Here is an assignment of probabilities for the outcomes: Time of day10 p.m. - 12:59 a.m.1 a.m. - 5:59 a.m.6 a.m. - 8:59 a.m.Probability0.0400.0330.144Time of Day9 a.m. - 12:59 p.m.1 p.m. - 3:59 p.m.4 p.m. - 6:59 p.m.Probability0.2340.208?Time of day7 p.m. - 9:59 p.m.Probability0.123 What is the probability that the chosen trip did not begin between 9 a.m. and 12:59 p.m.?

0.766

A company that offers courses to prepare students for the Graduate Management Admission Test (GMAT) has collected the following information about its customers: 20% are undergraduate students in business, 15% are undergraduate students in other fields of study, and 60% are college graduates who are currently employed. Choose a customer at random. Find the probability that the customer is not an undergraduate business student.

0.8

The Pew Research Center asked a random sample of 2024 adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data. Suppose we select one of the survey respondents at random. Age18-3435-5455+TotalType of cell phoneiPhone169171127467Android214189100503Other1342776431054Total5176378702024 What is the probability that the person is not age 18 to 34 and does not own an iPhone?

189+100+277+6432024=0.597

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data.Suppose we select one of the students at random. Grade4th grade5th grade6th gradeTotalMost importantGrades495069168Athletic24363898Popular19222869Total92108135335 What is the probability that the student is not a sixth-grader and did not rate good grades as important?

24+36+19+22335=0.301

A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits - clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jacks, queens, and kings are referred to as "face cards." Imagine that we shuffle the deck thoroughly and deal one card. Define events F: getting a face card and H: getting a heart. The two-way table summarizes the sample space for this chance process. CardFace cardNon-face cardTotalSuitHeart31013Non-heart93039Total124052 Find 𝑃(HC)

3952=0.75

In the game of Scrabble, each player begins by randomly selecting 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. We want to perform a simulation to determine the probability that a player will randomly select 7 vowels. How would you use a table of random digits to carry out this simulation?

Let the numbers 01-42 represent vowels, 43-98 represent consonants, and the 99 and 00 represent the blank tiles. Moving left to right across a row, look at pairs of digits until you have 7 unique numbers (no repetitions, because once you pull a tile from the bag you cannot pull it again). Record whether all 7 numbers are between 01 and 42 or not. Perform many trials of this simulation. Determine what percent of the trials had all 7 numbers between 01 and 42.

A professional tennis player claims to get 90% of her second serves in. In a recent match, the player missed 5 of her first 20 second serves. Is this a surprising result if the player's claim is true? Assume that the player has a 0.10 probability of missing each second serve. We want to carry out a simulation to estimate the probability that she would miss 5 or more of her first 20 second serves. The dotplot displays the number of second serves missed by the player out of the first 20 second serves in 100 simulated matches. Is there convincing evidence that the player misses more than 10% of her second serves? Why or why not?

No, there is not convincing evidence that the player misses more than 10% of her second serves. Assuming the player makes 90% of her second serves, there is only a 7% chance that she would miss 5 or more of her first 20 second serves.

A recent study reported that fewer than half of young adults turn off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS or 60 students at their school if they usually brush with the water off. In the sample, 27 students said "Yes." The dotplot shows the results of taking 200 SRSs of 60 students from a population in which the true population who brush with the water off is 0.50. Does the sample result (27 out of 60 students said "Yes") give convincing evidence that fewer than half of the school's students brush their teeth with the water off? Justify your response.

No, this is not convincing evidence that less than half the students at the school brush their teeth with the water off. In the simulation, 44 of the 200 samples yielded a sample proportion of 0.45 or less.

An airline reports that 85% of its flights arrive on time. To find the probability that a random sample of 4 of this airline's flights into LaGuardia Airport in New York City on the same night all arrive on time, can we multiply (0.85)(0.85)(0.85)(0.85)? Why or why not?

No. We cannot simply multiply probabilities together. If one flight is late, whatever is causing it to be late (weather, backed up airplanes, etc.) is likely to be affecting the three other flights. These events are not independent.

Abigail, Bobby, Carlos, DeAnna, and Emily go to the bagel shop for lunch every Thursday. Each time, they randomly pick 2 of the group to pay for lunch by drawing names from a hat. Find the probability that Carlos or DeAnna (or both) ends up paying for lunch.

P(Carlos or DeAnna) = 7/10 = 0.70

Canada has two official languages, English and French. Choose a Canadian at random and ask, "What is your mother tongue?" Here is a distribution of responses, combining many separate languages from the broad Asia/Pacific region. LanguageEnglishFrenchAsian/PacificOtherProbability0.630.220.060.09 What is the probability that the chosen person's mother tongue is one of Canada's official languages?

P(English or French) = 0.85

A professional tennis player claims to get 90% of her second serves in. In a recent match, the player missed 5 of her first 20 second serves. Is this a surprising result if the player's claim is true? Assume that the player has a 0.10 probability of missing each second serve. We want to carry out a simulation to estimate the probability that she would miss 5 or more of her first 20 second serves. The following steps describe how to use a random number generator to perform one trial of the simulation: Let 0 = miss the second serve and 1 to 9 = makes the second serve. Generate 20 random integers from 0 to 9 to simulate whether she makes or misses each of her first 20 second serves. Record the number of times she misses her second serve (the 0's). Do any of the simulation steps contain an error? If so, what is the correct procedure?

There are no errors.

In the game of Scrabble, each player begins by randomly selecting 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her 7 tiles and is surprised to discover that all of them are vowels. Suppose we performed a simulation to determine the probability that a player will randomly select 7 vowels and in 2 of the 1000 trials of the simulation, all 7 tiles were vowels. Does this give convincing evidence that Cait's bag of tiles was not well mixed?

Yes, because if the bag was well mixed, there is about a 0.2% chance of getting 7 tiles that are all vowels. Because it is very unlikely for this to happen by chance alone, we have reason to believe that the bag of tiles was not well mixed.

A recent study reported that fewer than half of young adults turn off the water while brushing their teeth. Is the same true for teenagers? To find out, a group of statistics students asked an SRS or 60 students at their school if they usually brush with the water off. In the sample, 27 students said "Yes." The dotplot shows the results of taking 200 SRSs of 60 students from a population in which the true population who brush with the water off is 0.50. Instead, if 18 of the students in the class's sample had said "Yes", would this give convincing evidence that fewer than half of the school's students brush their teeth with the water off? Justify your response.

Yes, this would be convincing evidence that less than half the students at the school brush their teeth with the water off. None of the 200 samples yielded a sample proportion of 0.30 or less.

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses. Choose a survey respondent at random. Define events G: good chance, M: male, and N: almost no chance. GenderFemaleMaleTotalOpinionAlmost no chance9698194Some chance but probably not426286712A 50-50 chance6967201416A good chance6637581421Almost certain4865971083Total236724594826 Find 𝑃(G∣M)P(G∣M). Interpret this value in context.

𝑃(G∣M)=7582459=0.308P(G∣M)=7582459=0.308. Given that the survey respondent is male, there is a 0.308 probability that the person responded "a good chance."

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses. Choose a survey respondent at random. Define events G: good chance, M: male, and N: almost no chance. GenderFemaleMaleTotalOpinionAlmost no chance9698194Some chance but probably not426286712A 50-50 chance6967201416A good chance6637581421Almost certain4865971083Total236724594826 Given that the chosen survey respondent didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.

𝑃(MC∣NC)=426+696+663+486712+1416+1421+1083=22714632=0.490

In January 2017, 52% of U.S. senators were Republicans and the rest were Democrats or Independents. Twenty-one percent of the senators were females, and 47% of the senators were male republicans. Suppose we select one of these senators at random. Define events R: is a Republican and M: is male. Consider the event that the randomly selected senator is a female Democrat or Independent. Write this event in symbolic form and find its probability.

𝑃(R𝐶∩M𝐶)P(RC∩MC)=0.16

Enzyme immunoassay (EIA) tests are used to screen blood specimens for the presence of antibodies to HIV, the virus that causes AIDS. Antibodies indicate the presence of the virus. The test is quite accurate but is not always correct. A false positive occurs when the test gives a positive result but no HIV antibodies are actually present in the blood. A false negative occurs when the test gives a negative result but HIV antibodies are present in the blood. Here are the approximate probabilities of positive and negative EIA outcomes when the blood tested does and does not actually contain antibodies to HIV: Suppose that 1% of a large population carries antibodies to HIV in their blood. Imagine choosing a person from this population at random. Test Result+-TruthAntibodies present0.99850.0015Antibodies absent0.00600.9940 If the person's EIA test is positive, what's the probability that the person has the HIV antibody?

𝑃(antibody∣positive)=(0.01)(0.9985)(0.01)(0.9985)+(0.99)(0.006)=0.627

Here is the distribution of the adjusted gross income (in thousands of dollars) reported on individual federal income tax returns in a recent year: Income<25<2525−4925−4950−9950−99100−499100−499≥500≥500Probability0.4310.2480.2150.1000.006 Given that a return shows an income of at least $50,000, what is the conditional probability that the income is at least $100,000?

𝑃(at least $100,000∣$50,000 or more)=0.1060.321=0.3302

Researchers carried out a survey of fourth-, fifth-, and sixth-grade students in Michigan. Students were asked whether good grades, athletic ability, or being popular was most important to them. The two-way table summarizes the survey data. Suppose we select one of the students at random. Grade4th grade5th grade6th gradeTotalMost importantGrades495069168Athletic24363898Popular19222869Total92108135335 Are the events "5th grade" and "athletic" independent? Justify your answer.

𝑃(athletic)=98335=0.293P(athletic)=98335=0.293 and 𝑃(athletic∣5th grade)=36108=0.333P(athletic∣5th grade)=36108=0.333Because these two probabilities are not equal, the events "athletic" and "5th grade" are not independent.

A standard deck of playing cards (with jokers removed) consists of 52 cards in four suits - clubs, diamonds, hearts, and spades. Each suit has 13 cards, with denominations ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. The jacks, queens, and kings are referred to as "face cards." Imagine that we shuffle the deck thoroughly and deal one card. The two-way table summarizes the sample space for this chance process based on whether or not the card is a face card and whether or not the card is a heart. CardFace cardNon-face cardTotalSuitHeart31013Non-heart93039Total124052 Are the events "heart" and "face card" independent? Justify your answer.

𝑃(heart)=0.25P(heart)=0.25 and 𝑃(heart∣face card)=0.25P(heart∣face card)=0.25Because these two probabilities are equal, the events "heart" and "face card" are independent.

The Pew Research Center asked a random sample of 2024 adult cell-phone owners from the United States their age and which type of cell phone they own: iPhone, Android, or other (including non-smartphones). The two-way table summarizes the data. Age18-3435-5455+TotalType of cell phoneiPhone169171127467Android214189100503Other1342776431054Total5176378702024 Are the events "iPhone" and "18-34" independent? Justify your answer.

𝑃(iPhone)=0.231P(iPhone)=0.231 and 𝑃(iPhone∣18−34)=0.327P(iPhone∣18−34)=0.327Because these two probabilities are not equal, the events "iPhone" and "18-34" are not independent.

A boy uses a homemade metal detector to look for valuable objects on a beach. The machine isn't perfect - it beeps for only 98% of the metal objects over which it passes, and it beeps for 4% of the nonmetallic objects over which it passes. Suppose that 25% of the objects that the machine passes over are metal. Choose an object from this beach at random. If the machine beeps when it passes over this object, find the probability that the boy has found a metal object.

𝑃(metal∣beeps)=(0.25)(0.98)(0.25)(0.98)+(0.75)(0.04)=0.891

Suppose you roll two fair, six-sided dice - one red and one blue. The figure shows the sample space of this chance process. Are the events "sum is 7" and "blue die shows a 4" independent? Justify your answer.

𝑃(sum of 7)=636=0.1667P(sum of 7)=636=0.1667 and 𝑃(sum of 7∣blue is 4)=16=0.1667P(sum of 7∣blue is 4)=16=0.1667Because these two probabilities are equal, the events "sum is 7" and "blue die shows a 4" are independent.

Suppose you roll two fair, six-sided dice - one red and one blue. The following figure shows the sample space of this chance process. Are the events "sum is 8" and "blue die shows a 4" independent? Justify your answer.

𝑃(sum of 8)=536=0.1398P(sum of 8)=536=0.1398 and 𝑃(sum of 8∣blue is 4)=16=0.1667P(sum of 8∣blue is 4)=16=0.1667Because these two probabilities are not equal, the events "sum is 8" and "blue die shows a 4" are not independent.

Select an adult at random. Define events D: person has earned a college degree, and T: person's career is teaching. Which of the following choices correctly ranks the following probabilities from smallest to largest?

𝑃(𝑇)<𝑃(𝑇∣𝐷)<𝑃(𝐷)<𝑃(𝐷∣𝑇)