Quiz 3 Chapter 5

 

1.   The table to the right lists probabilities for the corresponding numbers of girls in three births. What is the random​ variable, what are its possible​ values, and are its values​ numerical?

Choose the correct answer below.

1. The random variable is​ x, which is the number of girls in three births. The possible values of x are​ 0, 1,​ 2, and 3. The values of the random value x are not numerical.
2. The random variable is​ P(x), which is the probability of a number of girls in three births. The possible values of​ P(x) are 0.125 and 0.375. The values of the random value​ P(x) are numerical.
3. The random variable is​ P(x), which is the probability of a number of girls in three births. The possible values of​ P(x) are 0.125 and 0.375. The values of the random value​ P(x) are not numerical.
4. The random variable is​ x, which is the number of girls in three births. The possible values of x are​ 0, 1,​ 2, and 3. The values of the random value x are numerical.

 

2.  Is the random variable given in the accompanying table discrete or​ continuous? Explain.

The random variable given in the accompanying table is ________ because ___________________

 

3.     Five males with an​ X-linked genetic disorder have one child each. The random variable x is the number of children among the five who inherit the​ X-linked genetic disorder. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied.

Does the table show a probability​ distribution? Select all that apply.

​

1. Yes, the table shows a probability distribution.
2. ​No, the random variable x is categorical instead of numerical.
3. ​No, the sum of all the probabilities is not equal to 1.
4. ​No, the random variable​ x’s number values are not associated with probabilities.
5. ​No, not every probability is between 0 and 1 inclusive.

 

Find the mean of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

1. $\mu = \_\_\_\_\_ children$

   ​(Round to one decimal place as​ needed.)

2. The table does not show a probability distribution.

 

Find the standard deviation of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

 

1. $\sigma =$

   ______ ​child(ren) ​(Round to one decimal place as​ needed.)

2. The table does not show a probability distribution.

 

4.   Ted is not particularly creative. He uses the pickup line​ “If I could rearrange the​ alphabet, I’d put U and I​ together.” The random variable x is the number of women Ted approaches before encountering one who reacts positively. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied.

Does the table show a probability​ distribution? Select all that apply.
​

1. Yes, the table shows a probability distribution.
2. ​No, the sum of all the probabilities is not equal to 1.
3. ​No, the random variable x is categorical instead of numerical.
4. ​No, the random variable​ x’s number values are not associated with probabilities.
5. ​No, not every probability is between 0 and 1 inclusive.

Find the mean of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

1. $\mu =$

     _____ women ​(Round to one decimal place as​ needed.)

2. The table does not show a probability distribution.

Find the standard deviation of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

1. $\sigma = \_\_\_\_$

     women ​(Round to one decimal place as​ needed.)

2. The table does not show a probability distribution.

 

5   Groups of adults are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they would feel comfortable in a​ self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is​ given, find its mean and standard deviation. If a probability distribution is not​ given, identify the requirements that are not satisfied.

Does the table show a probability​ distribution? Select all that apply.

1. ​Yes, the table shows a probability distribution.
2. ​No, the sum of all the probabilities is not equal to 1.
3. ​No, not every probability is between 0 and 1 inclusive.
4. ​No, the random variable​ x’s number values are not associated with probabilities.
5. No, the random variable x is categorical instead of numerical.

 

Find the mean of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

1. $\mu =$

   ______ ​adult(s) ​(Round to one decimal place as​ needed.)

2. The table does not show a probability distribution.

 

Find the standard deviation of the random variable x. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

1. $\sigma =\_\_\_\_$

   ​adult(s) ​(Round to one decimal place as​ needed.) 

2. The table does not show a probability distribution.

 

 

6.  Refer to the accompanying​ table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Find the mean and standard deviation for the number of girls in 8 births.

The mean is

$\mu = \_\_\_\_$

​girl(s). ​(Round to one decimal place as​ needed.)

The standard deviation is

$\sigma = \_\_\_\_$

​girl(s). ​(Round to one decimal place as​ needed.)

 

7.   When playing roulette at a​ casino, a gambler is trying to decide whether to bet ​$15 on the number 27 or to bet ​$15 that the outcome is any one of the three
possibilities 00, 0, or 1.  The gambler knows that the expected value of the ​$15 bet for a single number is – $ 1.58. For the ​$15 bet that the outcome is 00, 0,  or 1​, 
there is a probability of

$\frac{3}{38}$

of making a net profit of ​$45 and a 

$\frac{35}{38}$

probability of losing ​$15

a. Find the expected value for the ​$15 bet that the outcome is 00, 0 or 1

b. Which bet is​ better: a ​$15 bet on the number 27 or a ​$15 bet that the outcome is any one of the numbers 00, 0, or 1 ? Why?

a. The expected value is ​$ _______ ​(Round to the nearest cent as​ needed.)

b. Since the expected value of the bet on the number 27 _________ than the expected value for the bet that the outcome is 00, 0, or 1​,  the bet on _________is better.

 

 

8.  There is a 0.9987 probability that a randomly selected 27 ​-year-old male lives through the year. A life insurance company charges ​$197 for insuring that the male will live through the year. If the male does not survive the​ year, the policy pays out ​$110,000 as a death benefit. Complete parts​ (a) through​ (c) below.

a. From the perspective of the 27 ​-year-old ​male, what are the monetary values corresponding to the two events of surviving the year and not​ surviving?

The value corresponding to surviving the year is ​__________

The value corresponding to not surviving the year is ________________ ​(Type integers or decimals. Do not​ round.)

b. If the 27 ​-year-old male purchases the​ policy, what is his expected​ value?

The expected value is _______. ​(Round to the nearest cent as​ needed.)

c. Can the insurance company expect to make a profit from many such​ policies? Why?

_______ , because the insurance company expects to make an average profit of _____ on every 27-year-old male it insures for 1 year.  (Round to the nearest cent as​ needed.)

 

9.   Fill in the blank.
A​ _______ variable is a variable that has a single numerical​ value, determined by​ chance, for each outcome of a procedure.

 

10.  Fill in the blank.
A​ _______ random variable has either a finite or a countable number of values.

 

11.   Fill in the blank.
A​ _______ random variable has infinitely many values associated with measurements.

 

12.  Fill in the blank.
In a probability​ histogram, there is a correspondence between​ _______.

 

13.  Fill in the blank.
The​ _______ of a discrete random variable represents the mean value of the outcomes.

 

14.  Based on a​ survey, assume that 48​%  of consumers are comfortable having drones deliver their purchases. Suppose that we want to find the probability that when six consumers are randomly​ selected, exactly four of them are comfortable with delivery by drones. Identify the values of​ n, x,​ p, and q.

The value of n is ____ . ​(Type an integer or a decimal. Do not​ round.)

The value of x is ____. ​(Type an integer or a decimal. Do not​ round.)

The value of p is _____.​(Type an integer or a decimal. Do not​ round.)

The value of q is ____ ​(Type an integer or a decimal. Do not​ round.)

 

15. Multiple-choice questions each have five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to three such questions.

a. Use the multiplication rule to find ​P(WWC), where C denotes a correct answer and W denotes a wrong answer.

​P(WWC = ______

b. Beginning with WWC, make a complete list of the different possible arrangements of one correct answer and two wrong answers​, then find the probability for each entry in the list.

​P(WWC​) – see above ​

P(WCW​ = ______

P(CWW) = ______ (Type exact​ answers.)

c. Based on the preceding​ results, what is the probability of getting exactly one correct answer when three guesses are​ made?

______ ​(Type an exact​ answer.)

 

 

16.  Assume that random guesses are made for six multiple choice questions on an SAT​ test, so that there are n = 6 ​trials, each with probability of success​ (correct) given by p = 0.2.  Find the indicated probability for the number of correct answers.

Find the probability that the number x of correct answers is fewer than 4

$P\left(X<4\right) =$

17. Assume that when adults with smartphones are randomly​ selected, 48​% use them in meetings or classes. If 6 adult smartphone users are randomly​ selected, find the probability that exactly 2 of them use their smartphones in meetings or classes.

The probability is

 

18.  Based on a​ poll, among adults who regret getting​ tattoos, 28​% say that they were too young when they got their tattoos. Assume that eight adults who regret getting tattoos are randomly​ selected, and find the indicated probability. Complete parts​ (a) through​ (d) below.

a. Find the probability that none of the selected adults say that they were too young to get tattoos.

________ ​(Round to four decimal places as​ needed.)

b. Find the probability that exactly one of the selected adults says that he or she was too young to get tattoos.

_________. ​(Round to four decimal places as​ needed.)

c. Find the probability that the number of selected adults saying they were too young is 0 or 1.

_________. ​(Round to four decimal places as​ needed.)

d. If we randomly select eight ​adults, is 1 a significantly low number who say that they were too young to get​ tattoos?

_________ because the probability that ________ of the selected adults say that they were too young is ___________ 0.05. 

 

 

19   A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 48 ​tablets, then accept the whole batch if there is only one or none that​ doesn’t meet the required specifications. If one shipment of 6000 aspirin tablets actually has a 3% rate of​ defects, what is the probability that this whole shipment will be​ accepted? Will almost all such shipments be​ accepted, or will many be​ rejected?

The probability that this whole shipment will be accepted is _______.  (Round to four decimal places as​ needed.)

The company will accept ______% of the shipments and will reject _____​% of the​ shipments, so ______________ (Round to two decimal places as​ needed.)

 

20 . Fill in the blank.
In the binomial probability​ formula, the variable x represents the​ _______.