Exam 1 STAT - HW 3
If you flip a fair coin 5 times, what is the probability of each of the following? b. getting all heads?
(1/2 heads)^(5 flips) = 0.03125
If you flip a fair coin 5 times, what is the probability of each of the following? a. getting all tails?
(1/2 tails)^(5 flips) = 0.03125
In a multiple choice exam, there are 7 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: b) she gets all of the questions right?
(1/4)^7 = 0.000061
In a multiple choice exam, there are 7 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: a) the first question she gets right is question number 3?
(3/4)^3 x (1/3)
In your sock drawer you have 4 blue, 4 gray, and 4 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing d) a green sock
0
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. f) What is the probability that a randomly chosen respondent is a moderate/liberal Republican given that they do not believe that the earth is warming?
0.02/0.12 = 0.1667
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. d) What is the probability that a randomly chosen respondent believes the earth is warming given that they are a conservative Republican?
0.06/0.20 = 0.3
Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 16% miss 2 days, and 23% miss 3 or more days due to sickness. c) What is the probability that a student chosen at random misses at least one day?
0.25 + 0.16 + 0.23 = 0.64
Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 16% miss 2 days, and 23% miss 3 or more days due to sickness. a) What is the probability that a student chosen at random doesn't miss any days of school due to sickness this year?
0.25 + 0.16 + 0.23 = 0.64 1 - 0.64 = 0.36
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. c) What is the probability that a randomly chosen respondent believes the earth is warming given that they are a liberal Democrat?
0.26/0.36 = 0.7222
Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 16% miss 2 days, and 23% miss 3 or more days due to sickness. b) What is the probability that a student chosen at random misses no more than one day?
0.36 + 0.25 = 0.61
In your sock drawer you have 4 blue, 4 gray, and 4 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing c) at least 1 black sock
0.5758
Drawing a face card (jack, queen, or king) and drawing a red card from a full deck of playing cards are mutually exclusive events.
False
If a fair coin is tossed many times and the last eight tosses are all heads, then the chance that the next toss will be heads is somewhat less than 50%.
False
Drawing a face card and drawing an ace from a full deck of playing cards are mutually exclusive events.
True
Identify each as a valid or invalid probability distribution 1. Distribution (b) is
Valid probability distribution
Identify each as a valid or invalid probability distribution 1. Distribution (e) is
Valid probability distribution
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. e) Does it appear that whether or not a respondent believes the earth is warming is independent of their party ideology?
belief in global warming and party ideology are dependent
In parts (a) and (b), identify whether the events are disjoint, independent, or neither (events cannot be both disjoint and independent). c) If two events can occur at the same time, they must be independent.
false
In parts (a) and (b), identify whether the events are disjoint, independent, or neither (events cannot be both disjoint and independent). a) You and a randomly selected student from your class both earn A's in this course.
independent
In parts (a) and (b), identify whether the events are disjoint, independent, or neither (events cannot be both disjoint and independent). b) You and your class partner both earn A's in this course.
neither
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. a) Are believing that the earth is warming and being a liberal Democrat mutually exclusive?
not mutually exclusive
A research poll asked 1599 Americans "From what you've read and heard, is there solid evidence that the average temperature on earth has been getting warmer over the past few decades, or not?". The table below shows the distribution of responses by party and ideology, where the counts have been replaced with relative frequencies. b) What is the probability that a randomly chosen respondent believes the earth is warming or is a liberal Democrat?
0.58 are believers 0.36 Liberal Democrats 0.26 are liberals democrats and believers 0.58 + 0.36 - 0.26 = 0.68
Suppose 82% of people like peanut butter, 85% like jelly, and 82% like both. Given that a randomly sampled person likes peanut butter, what's the probability that he also likes jelly?
0.82/0.82 = 1
If you flip a fair coin 5 times, what is the probability of each of the following? c. getting at least one tails?
1 - (1/2)^5 = 0.9688
In a multiple choice exam, there are 7 questions and 4 choices for each question (a, b, c, d). Nancy has not studied for the exam at all and decides to randomly guess the answers. What is the probability that: c) she gets at least one question right?
1 - (3/4)^7 = 0.8665
A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 32% of respondents identified as Independent, 21% identified as swing voters, and 11% identified as both. Draw a Venn diagram summarizing the variables and their associated probabilities. d) What percent of voters are neither Independent nor swing voters?
1 - 0.42 = 58%
Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. a. If the proportion of heads is larger than 0.60, you win $1.
10 tosses
Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. d. If the proportion of heads is smaller than 0.30, you win $1.
10 tosses
Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. b. If the proportion of heads is larger than 0.40, you win $1.
100 tosses
Below are four versions of the same game. Your archnemesis gets to pick the version of the game, and then you get to choose how many times to flip a coin: 10 times or 100 times. Identify how many coin flips you should choose for each version of the game. It costs $1 to play each game. c. If the proportion of heads is between 0.40 and 0.60, you win $1.
100 tosses
Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 201 Scandinavian men and their female partners. The table below summarizes the results (rows represent male eye color while columns represent female eye color). For simplicity, we only include heterosexual relationships in this exercise. a) What is the probability that a randomly chosen male respondent or his partner has blue eyes?
108/201 + 89/201 - 62/201 = 0.6716
Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 201 Scandinavian men and their female partners. The table below summarizes the results (rows represent male eye color while columns represent female eye color). For simplicity, we only include heterosexual relationships in this exercise. d) What is the probability of a randomly chosen male respondent with green eyes having a partner with blue eyes?
13/40 = 0.3250
At a university, 14% of students smoke a) Calculate the expected number of smokers in a random sample of 140 students from this university
14/100 x 140 = 19.6
Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 201 Scandinavian men and their female partners. The table below summarizes the results (rows represent male eye color while columns represent female eye color). For simplicity, we only include heterosexual relationships in this exercise. c) What is the probability that a randomly chosen male respondent with brown eyes has a partner with blue eyes?
14/53 = 0.2642
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. You watch a roulette wheel spin 10 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin?
18 red / 38 total slots = 0.4737
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. You watch a roulette wheel spin 180 consecutive times and the ball lands on a red slot each time. What is the probability that the ball will land on a red slot on the next spin?
18 red / 38 total slots = 0.4737
The histogram shown below represents the weights (in kg) of 47 female and 97 male cats. Approximately % of these cats weigh between 2.5 and 2.75kg
20
The histogram shown below represents the weights (in kg) of 47 female and 97 male cats. Approximately % of these cats weigh between 2.75 and 3.5kg
27
The histogram shown below represents the weights (in kg) of 47 female and 97 male cats. Approximately % of these cats weigh less than 2.5kg
33
Imagine you have an urn containing 4 red, 4 blue, and 2 orange marbles in it. a) What is the probability that the first marble you draw is blue?
4/10 = 0.4
Imagine you have an urn containing 4 red, 4 blue, and 2 orange marbles in it. b) Suppose you drew a blue marble in the first draw. If drawing with replacement, what is the probability of drawing a blue marble in the second draw?
4/10 = 0.4
Imagine you have an urn containing 4 red, 4 blue, and 2 orange marbles in it. c) Suppose you instead drew an orange marble in the first draw. If drawing with replacement, what is the probability of drawing a blue marble in the second draw?
4/10 = 0.4
Imagine you have an urn containing 4 red, 4 blue, and 2 orange marbles in it. d) If drawing with replacement, what is the probability of drawing two blue marbles in a row?
4/10 x 4/10 = 0.16
In your sock drawer you have 4 blue, 4 gray, and 4 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing a) 2 blue socks
4/12 x 3/11 = 0.09091
The relative frequency table below displays the distribution of annual total personal income (in 2009 inflation-adjusted dollars) for a representative sample of 96,420,486 Americans. These data come from the American Community Survey for 2005-2009. This sample is comprised of 59% males and 41% females. b) What is the probability that a randomly chosen US resident makes less than $50,000 per year and is female?
41/100 x 62.2/100 = 0.2550
The Behavioral Risk Factor Surveillance System (BRFSS) is an annual telephone survey designed to identify risk factors in the adult population and report emerging health trends. The following table summarizes two variables: health status (excellent, very good, good, fair, poor) and health coverage (no, yes), which describes whether each respondent had health insurance. a) If we draw one individual at random, what is the probability that the respondent has excellent health and doesn't have health coverage?
459/20000
The relative frequency table below displays the distribution of annual total personal income (in 2009 inflation-adjusted dollars) for a representative sample of 96,420,486 Americans. These data come from the American Community Survey for 2005-2009. This sample is comprised of 59% males and 41% females. a) What is the probability that a randomly chosen US resident makes less than $50,000 per year?
62.2/100 = 0.6220
Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 201 Scandinavian men and their female partners. The table below summarizes the results (rows represent male eye color while columns represent female eye color). For simplicity, we only include heterosexual relationships in this exercise. b) What is the probability that a randomly chosen male respondent with blue eyes has a partner with blue eyes?
62/108 = 0.5741
In your sock drawer you have 4 blue, 4 gray, and 4 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing b) no gray socks
8/12 x 7/11 = 0.4242
Identify each as a valid or invalid probability distribution 1. Distribution (a) is
Invalid probability distribution
Identify each as a valid or invalid probability distribution 1. Distribution (c) is
Invalid probability distribution
Identify each as a valid or invalid probability distribution 1. Distribution (d) is
Invalid probability distribution
Identify each as a valid or invalid probability distribution 1. Distribution (f) is
Invalid probability distribution
A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 32% of respondents identified as Independent, 21% identified as swing voters, and 11% identified as both. Draw a Venn diagram summarizing the variables and their associated probabilities. a) Are being Independent and being a swing voter disjoint, i.e. mutually exclusive?
No
Assortative mating is a nonrandom mating pattern where individuals with similar genotypes and/or phenotypes mate with one another more frequently than what would be expected under a random mating pattern. Researchers studying this topic collected data on eye colors of 201 Scandinavian men and their female partners. The table below summarizes the results (rows represent male eye color while columns represent female eye color). For simplicity, we only include heterosexual relationships in this exercise. e) Does it appear that the eye colors of male respondents and their partners are independent?
No, it is much more likely for a male with blue eyes to have a blue-eyed partner than it is for a male with any other eye color
At a university, 14% of students smoke b) The university gym opens at 9 am on Saturday mornings. One Saturday morning at 8:55 am there are 21 students outside the gym waiting for it to open. Should you use the same approach from part (a) to calculate the expected number of smokers among these 21 students?
No, it is unlikely that smoking habits and waking up early to go to the gym on Saturday are independent
A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 32% of respondents identified as Independent, 21% identified as swing voters, and 11% identified as both. Draw a Venn diagram summarizing the variables and their associated probabilities. e) Is the event that someone is a swing voter independent of the event that someone is a political Independent?
No, they are dependent
In your sock drawer you have 4 blue, 4 gray, and 4 black socks. Half asleep one morning you grab 2 socks at random and put them on. Find the probability you end up wearing e) matching socks
P(1st blue and 2nd blue or 1st gray and 2nd gray, or 1st black and 2nd black) = P(1st blue)*P(2nd blue | 1st blue) + P(1st gray)*P(2nd gray|1st gray) + P(1st black)*P(2nd black|1st black) = (4/12 x 3/11) + (4/12x3/11) + (4/12x3/11) = 0.2727
The Behavioral Risk Factor Surveillance System (BRFSS) is an annual telephone survey designed to identify risk factors in the adult population and report emerging health trends. The following table summarizes two variables: health status (excellent, very good, good, fair, poor) and health coverage (no, yes), which describes whether each respondent had health insurance. If we draw one individual at random, what is the probability that the respondent has excellent health or doesn't have health coverage?
P(A or B) = P(A)+P(B)-P(A and B) = 4657/20000 + 2524/2000 - 459/20000
A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 32% of respondents identified as Independent, 21% identified as swing voters, and 11% identified as both. Draw a Venn diagram summarizing the variables and their associated probabilities. c) What percent of voters are Independent or swing voters?
P(I or S) = P(I) + P(S) - P(I n S) = 0.32 + 0.21 - 0.11 = 42%
Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 16% miss 2 days, and 23% miss 3 or more days due to sickness. e) If a parent has two kids at a DeKalb County elementary school, what is the probability that both kids will miss some school, i.e. at least one day? Assume the two kids' attendance records are independent.
Probability of both kids miss any school = Probabily 1st kid will miss * Probability second kid will miss = 0.64 x 0.64 = 0.4096
Data collected at elementary schools in DeKalb County, GA suggest that each year roughly 25% of students miss exactly one day of school, 16% miss 2 days, and 23% miss 3 or more days due to sickness. d) If a parent has two kids at a DeKalb County elementary school, what is the probability that neither kid will miss any school? Assume the two kids' attendance records are independent.
Probability of neither kid miss any school = Probabily 1st kid will not miss * Probability second kid will not miss = 0.36 x 0.36 = 0.1296
A 2012 Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. 32% of respondents identified as Independent, 21% identified as swing voters, and 11% identified as both. Draw a Venn diagram summarizing the variables and their associated probabilities. b) What percent of voters are Independent but not swing voters?
Since A and B complement means the outcomes where A happens but B does not happens P(A and B^c) = P(A)-P(A and B) = 0.32 - 0.11 = 0.21
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) Andy's expected profit per game is:
X =winning Sum_ x(PX=x) = (0x36/52) + (3x12/52) + (5x3/52) + ((5+25)x1/52) =1.5577 Since ace club receives extra 20
Imagine you have an urn containing 4 red, 4 blue, and 2 orange marbles in it. e) When drawing with replacement, are the draws independent?
Yes, they are independent
