Test 2
A golf ball is selected at random from a golf bag. If the golf bag contains 5 black balls, 9 yellow balls, and 10 brown balls, find the probability of the following event. The golf ball is black or yellow.
(5 + 9)/24 .583
Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not alike?
(52 * 1056)/ 2598960 =.0211
If r=_______, then a perfect negative linear relation exists between the two quantitative variables.
-1
Suppose you toss a coin 100 times and get 77 heads and 23 tails. Based on these results, what is the probability that the next flip results in a tail?
.23
Suppose that E and F are two events and that P(E)=0.4 and P(F|E)=0.3. What is P(E and F)?
.3*.4 = .12
Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Suppose the outcomes are equally likely. Compute the probability of the event E="an even number less than 9."
.4
Suppose that E and F are two events and that P(E and F)=0.4 and P(E)=0.5. What is P(F|E)?
.4/.5 = .8
Let the sample space be S={1, 2, 3, 4, 5, 6, 7, 8}. Suppose the outcomes are equally likely. Compute the probability of the event E="an odd number."
.5
Suppose that events E and F are independent, P(E)=0.7, and P(F)=0.9. What is the P(E and F)
.63 .7*.9 = .63
If P(E)=0.50, P(E or F)=0.70, and P(E and F)=0.05, find P(F).
.7 + .05 - .5
What is the probability of an event that is impossible? Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible? What is the probability of an event that is impossible?
0
A license plate is to consist of 3 digits followed by 5 uppercase letters. Determine the number of different license plates possible if the first and second digits must be odd, and repetition is not permitted.
1,262,976,000
5!
120
Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck?
13C1 * 4C3 = 52
A golf-course architect has six linden trees, five white birch trees, and three bald cypress trees to plant in a row along a fairway. In how many ways can the landscaper plant the trees in a row, assuming that the trees are evenly spaced?
14! / 6! * 5! *3! 168,168
The probability that between
15<x<17
(b) Suppose a woman in the country aged 25 years or older is randomly selected. What is the probability she has a Bachelor's Degree and has never married? Interpret this probability.
18% * 20.7% .18 * .207 .0037 This probability means that if 100 women in the country aged 25 years or older were randomly selected, one could expect about 4 of them to have a Bachelor's Degree and never have married.
If a person rolls a six-sided die and then draws a playing card and checks its color, describe the sample space of possible outcomes using 1, 2, 3, 4, 5, 6 for the die outcomes and B, R for the card outcomes. (Make sure your answers reflect the order stated.)
1B, 1R, 2B, 2R, 3B, 3R, 4B, 4R, 5B, 5R, 6B , 6R
A man has two shirts and four ties. Assuming that they all match, how many different shirt-and-tie combinations can he wear?
2*4 = 8
A license plate is to consist of 4 digits followed by 5 uppercase letters. Determine the number of different license plates possible if the first and second digits must be odd, and repetition is not permitted.
5*4*8*7*26*25*24*23*22 =8,840,832,000
A salesperson must travel to six cities to promote a new marketing campaign. How many different trips are possible if any route between cities is possible?
6! 720
Suppose Aaron is going to build a playlist that contains 6 songs. In how many ways can Aaron arrange the 6 songs on the playlist?
6! 720
6P1
6!/(6!-1!) =6
9C6
9!/6!(9!-6!) 9!/3! 84
The probability that fewer than
<
The probability that exactly
=
The probability that at least
>
What is a random variable?
A random variable is a numerical measure of the outcome of a probability experiment.
What is a residual? What does it mean when a residual is positive?
A residual is the difference between an observed value of the response variable y and the predicted value of y. If it is positive, then the observed value is greater than the predicted value.
The word or in probability implies that we use the _______ rule
Addition
Which type of compound event is generally associated with multiplication? Which is generally associated with addition?
An 'AND' compound event is generally associated with multiplication; an 'OR' compound event is generally associated with addition.
What does it mean for an event to be unusual? Why should the cutoff for identifying unusual events not always be 0.05?
An event is unusual if it has a low probability of occurring. The choice of a cutoff should consider the context of the problem.
Describe what an unusual event is. Should the same cutoff always be used to identify unusual events? Why or why not?
An event is unusual if it has a low probability of occurring. The same cutoff should not always be used to identify unusual events. Selecting a cutoff is subjective and should take into account the consequences of incorrectly identifying an event as unusual.
Describe how the value of n affects the shape of the binomial probability histogram.
As n increases, the binomial distribution becomes more bell shaped.
Which of the following interpretations of the mean is correct?
As the number of experiments increases, the mean of the observations will approach the mean of the random variable.
Explain the Law of Large Numbers. How does this law apply to gambling casinos?
As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome. This applies to casinos because they are able to make a profit in the long run because they have a small statistical advantage in each game.
The remaining 2 cards must be different from the 3 chosen and different from each other. After selecting the three of a kind, there are 12 different ranks of cards remaining in the deck that can be chosen. Of the 12 ranks remaining, the player chooses 2 of them and then selects one of the 4 cards in each of the two chosen ranks. How many ways can the player select the remaining 2 cards?
Choose 2 from 12 12c2 =66 Choose 2 cards from each of 4 rank. 4c2 * 4c2 =16 66 * 16 =1056
The probability of having seven girls in an seven-child family is 0.0078125.
Classical method
D, J, R , C work for a publishing company. The company wants to send two employees to a statistics conference. To be fair, the company decides that the two individuals who get to attend will have their names randomly drawn from a hat. (a) Determine the sample space of the experiment. That is, list all possible simple random samples of size n=2.
DJ, DR, DC, JR, JC, RC
A probability experiment is conducted in which the sample space of the experiment is S={2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, event E={4, 5, 6, 7, 8, 9} and event G={10, 11, 12, 13}. Assume that each outcome is equally likely. List the outcomes in E and G. Are E and G mutually exclusive? List the outcomes in E and G. Choose the correct answer below.
E and G = {}
What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success p?
E(X)=np
What are the two requirements for a discrete probability distribution?
EP(x) = 1 0 (< or equal) P(x) (< or equal) 1
Explain what each point on the least-squares regression line represents.
Each point on the least-squares regression line represents the predicted y-value at the corresponding value of x.
A binomial experiment is performed a fixed number of times. What is each repetition of the experiment called?
Each repetition of the experiment is called a trial.
State the criteria for a binomial probability experiment. Choose the correct answer below. Select all that apply.
Each trial has two possible mutually exclusive outcomes: success and failure. The trials are independent. The experiment consists of a fixed number, n, of trials. The probability of success, p, remains constant for each trial of the experiment
A probability experiment is conducted in which the sample space of the experiment is S={10,11,12,13,14,15,16,17,18,19,20,21}. Let event E={12,13,14,15,16,17,18,19,20}. Assume each outcome is equally likely. List the outcomes in Ec. Find PEc List the outcomes in Ec. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. Ec = P(Ec) =
Ec = {10,11,21} pEc = .25
What method of assigning probabilities to a simple event uses relative frequencies?
Empirical
On the basis of a survey of 1000 families with seven children, the probability of a family having seven girls is 0.0059.
Empirical method
On the basis of clinical trials, the probability of efficacy of a new drug is 0.74.
Empirical method
The notation P(F E) means the probability of event __ given event __
F E
True or false: Correlation implies causation.
False (Often times in observational studies, we cannot conclude two correlated variables have a causal relationship. The presence of a lurking variable that is related to both the explanatory variable and the response variable can make the two variables correlated without having a causal relation.)
A player is dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering questions a through d. a) How many ways can 5 cards be selected from a 52-card deck?
How many ways can 5 cards be selected from a 52-card deck? 52C5 2598960
If the linear correlation between two variables is negative, what can be said about the slope of the regression line?
Negative
Will the following variables have positive correlation, negative correlation, or no correlation? interest rates on car loans and number of cars sold
Negative
Suppose that a probability is approximated to be zero based on empirical results. Does this mean that the event is impossible?
No
The linear correlation between violent crime rate and percentage of the population that has a cell phone is −0.918 for years since 1995. Do you believe that increasing the percentage of the population that has a cell phone will decrease the violent crime rate? What might be a lurking variable between percentage of the population with a cell phone and violent crime rate? Will increasing the percentage of the population that has a cell phone decrease the violent crime rate? Choose the best option below.
No
The probability that a randomly selected individual in a country earns more than $75,000 per year is 8.5%. The probability that a randomly selected individual in the country earns more than $75,000 per year, given that the individual has earned a bachelor's degree, is 21.5%. Are the events "earn more than $75,000 per year" and "earned a bachelor's degree" independent?
No
What does it mean if r=0?
No linear relationship exists between the variables.
If events E and F are disjoint and the events F and G are disjoint, must the events E and G necessarily be disjoint? Give an example to illustrate your opinion.
No, events E and G are not necessarily disjoint. For example, E={0,1,2}, F={3,4,5}, and G={2,6,7} show that E and F are disjoint events, F and G are disjoint events, and E and G are events that are not disjoint.
Determine whether the distribution is a discrete probability distribution.
No, because each probability is not between 0 and 1, inclusive.
Which of the following interpretations of the mean is correct?
Over the course of many games, one would expect the mean number of hits per game to be the mean of the random variable.
Find the probability of the indicated event if P(E)=0.35 and P(F)=0.45. Find P(E and F) if P(E or F)=0.55
P(E and F)= .25 .35 + .45 - .55 = .25
If E and F are not disjoint events, then P(E or F)=________.
P(E) + P(F) - P(E and F)
If E and F are disjoint events, then P(E or F)=
P(E)+P(F)
According to a sports analyst, the probability that a football team will win the next game is 0.41.
Subjective method
Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is linear, determine whether it indicates a positive or negative association between the variables. Use this information to answer the following. Do the two variables have a linear relationship?
The data points do not have a linear relationship because they do not lie mainly in a straight line.
Determine whether the scatter diagram indicates that a linear relation may exist between the two variables. If the relation is linear, determine whether it indicates a positive or negative association between the variables Do the two variables have a linear relationship?
The data points have a linear relationship because they lie mainly in a straight line.
Describe the difference between classical and empirical probability.
The empirical method obtains an approximate empirical probability of an event by conducting a probability experiment. The classical method of computing probabilities does not require that a probability experiment actually be performed. Rather, it relies on counting techniques, and requires equally likely outcomes.
Bob is asked to construct a probability model for rolling a pair of fair dice. He lists the outcomes as 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Because there are 11 outcomes, he reasoned, the probability of rolling a three must be 111. What is wrong with Bob's reasoning?
The experiment does not have equally likely outcomes.
Identify the statements that explain why this is a binomial experiment. Select all that apply.
The experiment is performed a fixed number of times. The trials are independent. There are two mutually exclusive outcomes, success or failure. The probability of success is the same for each trial of the experiment.
What might the lawyer of a defendant from this minority race argue?
The number of minorities on the jury is unusually low, given the composition of the population from which it came.
Which of the following numbers could be the probability of an event? 0.01, −0.55, 0.28, 1.2, 0, 1
The numbers that could be a probability of an event are 0, 0.01 , .028 , 1
Find the probability P(Ec) if P(E)=0.46.
The probability P(Ec) is .54
n a certain card game, the probability that a player is dealt a particular hand is 0.28. Explain what this probability means. If you play this card game 100 times, will you be dealt this hand exactly 28 times? Why or why not?
The probability 0.28 means that approximately 28 out of every 100 dealt ands will be that particular hand. No, you will not be dealt this hand exactly 28 times since the probability refers to what is expected in the long-term, not short-term.
According to a government statistics department, 20.7% of women in a country aged 25 years or older have a Bachelor's Degree; 15.8% of women in the country aged 25 years or older have never married; among women in the country aged 25 years or older who have never married, 23.6% have a Bachelor's Degree; and among women in the country aged 25 years or older who have a Bachelor's Degree, 18.0% have never married. Complete parts (a) and (b) below. Are the events "have a Bachelor's Degree" and "never married" independent? Explain.
The probability of the event "have a Bachelor's Degree" is affected by the occurrence of the event "never married", and the probability of the event "never married" is affected by the occurrence of the event "have a Bachelor's Degree", so the events are not independent.
Find the probability P(E or F) if E and F are mutually exclusive, P(E)=0.25, and P(F)=0.51
The probability P(E or F) is .76 .25 + .51 - 0 = .76
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. Is the distance a baseball travels in the air after being hit discrete or continuous?
The random variable is continuous. The possible values are d>0
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. Is the time it takes to fly from City A to City B discrete or continuous?
The random variable is continuous. The possible values are t>0
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. Is the time required to download a file from the Internet discrete or continuous?
The random variable is continuous. The possible values are t>0.
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. Is the number of hits to a website in a day discrete or continuous
The random variable is discrete. The possible values are x = 0,1,2,3...
Determine whether the random variable is discrete or continuous. In each case, state the possible values of the random variable. The number of light bulbs that burn out in the next week in a room with 14 bulbs.
The random variable is discrete. The possible values are x=0, 1, 2,..., 14.
If the relationship is linear do the variables have a positive or negative association?
The relationship is not linear.
In a certain game of chance, a wheel consists of 38 slots numbered 00, 0, 1, 2,..., 36. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. Complete parts (a) through (c) below.
The sample space is {00, 0, 1, 2,...,36}
Do the two variables have a positive or a negative association?
The two variables have a positive association.
What does it mean to say that two variables are negatively associated?
There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable decreases.
What does it mean to say that two variables are positively associated?
There is a linear relationship between the variables, and whenever the value of one variable increases, the value of the other variable increases.
A poll reported that 63% of adults were satisfied with the job the major airlines were doing. Suppose 20 adults are selected at random and the number who are satisfied is recorded Explain why this is a binomial experiment. Choose the correct answer below.
This is a binomial experiment because there are two mutually exclusive outcomes for each trial, there is a fixed number of trials, the outcome of one trial does not affect the outcome of another, and the probability of success is the same for each trial.
Determine if the following statement is true or false. In the binomial probability distribution function, nCx represents the number of ways of obtaining x successes in n trials
True
Determine if the following statement is true or false. Probability is a measure of the likelihood of a random phenomenon or chance behavior.
True
In a combination problem, order is not important.
True
Is the statement below true or false? The least-squares regression line always travels through the point (xbar, ybar)
True
True or False: In a probability model, the sum of the probabilities of all outcomes must equal 1.
True
When can the Empirical Rule be used to identify unusual results in a binomial experiment? Why can the Empirical Rule be used to identify results in a binomial experiment?
When the binomial distribution is approximately bell shaped, about 95% of the outcomes will be in the interval from μ−2σ to μ+2σ. The Empirical Rule can be used to identify results in binomial experiments when np(1−p)≥10
What does it mean to say that the linear correlation coefficient between two variables equals 1? What would the scatter diagram look like?
When the linear correlation coefficient is 1, there is a perfect positive linear relation between the two variables. The scatter diagram would contain points that all lie on a line with a positive slope.
Would it be unusual to observe 360 smokers who started smoking before turning 18 years old in a random sample of 400 adult smokers?
Yes, because 360 is greater than μ+2σ.
You suspect a 6-sided die to be loaded and conduct a probability experiment by rolling the die 400 times. The outcome of the experiment is listed in the following table. Do you think the die is loaded? Why?
Yes, because two of the values have a higher probability of occurring than expected under the assumption of equally likely outcomes.
List all the combinations of four objects a, b, c, and d taken three at a time.
abc, abd, acd, bcd
A ________ is an arrangement of r objects chosen from n distinct objects without repetition and without regard to order.
combination
A(n) is any collection of outcomes from a probability experiment.
event
In probability, a(n) ________ is any process that can be repeated in which the results are uncertain.
experiment
Two events E and F are ________ if the occurrence of event E in a probability experiment does not affect the probability of event F.
independent
List all the permutations of four objects m, l, n, and k taken two at a time without repetition. Choose the correct answer below.
ml, mn, mk, lm, ln, lk, nm, nl, nk, km, kl, kn
The word and in probability implies that we use the ________ rule.
multiplication
Fewer than half of 18 individuals covering their mouth would be surprising because the probability of observing fewer than half covering their mouth when sneezing is 0.0089,which is an unusual event.
n = 18 p = .267 pC = 1 - .267 ... .733 PROB HALF OF 18 = 9 In stat crunch, put in value of n, pC, and 9
In a certain lottery, an urn contains balls numbered 1 to 35. From this urn, 6 balls are chosen randomly, without replacement. For a $1 bet, a player chooses one set of six numbers. To win, all six numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning this lottery with one ticket?
nCr n = 35 r = 6 35!/(35!-6!)6! = x 1/x = 6.16 * 10^-7
How many different simple random samples of size 5 can be obtained from a population whose size is 44?
nCr n = 44 r = 5 44!/5!(44!-5!) 1,086,008
Four members from a 18-person committee are to be selected randomly to serve as chairperson, vice-chairperson, secretary, and treasurer. The first person selected is the chairperson; the second, the vice-chairperson; the third, the secretary; and the fourth, the treasurer. How many different leadership structures are possible?
nPr n = 18 r = 4 (Chairperson, VP, S , T) 18!/(18!-4!) = 73440
A(n) _________ is an ordered arrangement of r objects chosen from n distinct objects without repetition.
permutation
What might be a lurking variable between percentage of the population with a cell phone and violent crime rate?
the economy
According to a center for disease control, the probability that a randomly selected person has hearing problems is 0.157. The probability that a randomly selected person has vision problems is 0.095. Can we compute the probability of randomly selecting a person who has hearing problems or vision problems by adding these probabilities? Why or why not?
No, because hearing and vision problems are not mutually exclusive. So, some people have both hearing and vision problems. These people would be included twice in the probability.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. A baseball pitcher who strikes out 27% of his batters is asked to throw pitches until he strikes out a batter. The number of pitches attempted is recorded
No, because the experiment is not performed a fixed number of times.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Five cards are selected from a standard 52-card deck without replacement. The number of clubs selected is recorded
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. Four cards are selected from a standard 52-card deck without replacement. The number of threes selected is recorded.
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Determine if the following probability experiment represents a binomial experiment. A random sample of 80 middle school students is obtained, and the individuals selected are asked to state their hair length.
No, this probability experiment does not represent a binomial experiment because the variable is continuous, and there are not two mutually exclusive outcomes.
Find the probability of the indicated event if P(E)=0.35 and P(F)=0.50. Find P(E or F) if P(E and F)=0.05
P(E or F)= .8 .35 + .50 - .05 = .8
Are E and G mutually exclusive?
Yes, because the events E and G have no outcomes in common.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 120 randomly selected individuals, with the number of individuals responding favorably recorded
Yes, because the experiment satisfies all the criteria for a binomial experiment.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An experimental drug is administered to 30 randomly selected individuals, with the number of individuals responding favorably recorded.
Yes, because the experiment satisfies all the criteria for a binomial experiment.
Determine whether the following probability experiment represents a binomial experiment and explain the reason for your answer. An investor randomly purchases 6 stocks listed on a stock exchange. Historically, the probability that a stock listed on this exchange will increase in value over the course of a year is 40%. The number of stocks that increase in value is recorded.
Yes, because the experiment satisfies all the criteria for a binomial experiment.
Determine whether the distribution is a discrete probability distribution.
Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and 1, inclusive.
The following table shows the distribution of murders by type of weapon for murder cases in a particular country over the past 12 years. Complete parts (a) through (e). Is the given table a probability model? Why or why not?
Yes; the rules required for a probability model are both met.
The factorial symbol, n!, is defined as n!=_______ and 0!=_______.
n!=n(n−1)•...•3•2•1 and 0!=1.
