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?

** 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. **

*The variable x is a random variable because its values depend on
chance. The possible values of x are 0, 1, 2, and 3. Because these
values are numbers, x is a numerical variable.*

Determine whether the following value is a continuous random variable, discrete random variable, or not a random variable.

- The exact time it takes to evaluate 67 + 29
- The "yes" or "no" response to a survey question
- The weight of a hamburger
- The amount of rainfall in a country in a year
- The time required to upload a file to the Internet
- The number of people with blood type B in a random sample of 29 people

**a.** It is a continuous random variable.

**b.** It is not a random variable.

**c.** It is a continuous random variable.

**d.** It is a continuous random variable.

**e.** It is a continuous random variable.

**f.** It is a discrete random variable.

Determine whether the following value is a continuous random variable, discrete random variable, or not a random variable.

- The number of runs scored during a baseball game
- The last book a person in City A read
- The number of pigeons in a country
- The distance a football travels in the air after being thrown
- The number of hits to a website in a week
- The number of home runs in a baseball game

**a.** It is a discrete random variable.

**b.** It is not a random variable.

**c.** It is a discrete random variable.

**d.** It is a continuous random variable.

**e.** It is a discrete random variable.

**f.** It is a discrete random variable.

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 girls Ted approaches before encountering one who reacts positively.

Determine whether the table describes a probability distribution. If it does, find its mean and standard deviation.

The table is not a probability distribution.

*Σ P(x) = 0.902 ≠ 1.00*

Groups of people aged 15-65 are randomly selected and arranged in groups of six. In the accompanying table, the random variable x is the number in the group who say that their family and/or partner contribute most to their happiness (based on a survey).

If the table is a probability distribution, what is its mean?

Its mean is ** 4.5**.

Its standard deviation is ** 1.1**.

The accompanying data table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children.

Use the range rule of thumb to identify a range of values containing the usual numbers of girls in 10 births.

Based on the result, is 1 girl in 10 births an unusually low number of girls? Explain.

The maximum usual value is ** 8.4**.

*The range rule of thumb for the maximum usual value is given by
the formula below, where μ and σ are the mean and standard
deviation, respectively, for a probability distribution.*

*maximum usual value = μ + 2σ*

*= 5.034 + 2(1.696126175)*

*= 8.42625235*

The minimum usual value is **1.6**.

*minimum u* *sual value = μ – 2σ*

*= 5.034 – 2(1.696126175)*

*= 1.64174765*

**Yes, 1 girl is an unusually low number of girls, because 1
girl is outside of the range of usual values.**

The accompanying table describes results of roadworthiness tests of cars that are 3 years old. The random variable x represents the numbers of cars that failed among six that were tested for roadworthiness.

Find the mean and standard deviation for the number of cars that failed among the six cars that are tested.

* μ* =

**1.3**

* σ* =

**1.2**

Assume that 12 jurors are selected from a population in which 80% of the people are Mexican-Americans. The random variable x is the number of Mexican-Americans on the jury.

- Find the probability of exactly 8 Mexican-Americans among 12 jurors.
- Find the probability of 8 or fewer Mexican-Americans among 12 jurors.
- Which probability is relevant for determining whether 8 jurors among 12 is unusually low: the result from part (a) or part (b)?
- Is 8 an unusually low number of Mexican-Americans among 12 jurors? Why or why not?

**a.** P(8) = ** 0.133**

**b.** The probability of 8 or fewer Mexican-Americans
among 12 jurors is **0.206**.

*(0 + 0 + 0 + 0 + 0.001 + 0.003 + 0.016 + 0.053 + 0.133) = 0.206*

**c.** The result from part (b), because it measures the
probability of 8 or fewer successes.

**d.** No, because the relevant probability is greater
than 0.05.

Assume that 12 jurors are selected from a population in which 80% of the people are Mexican-Americans. The random variable x is the number of Mexican-Americans on the jury.

- Find the probability of exactly 5 Mexican-Americans among 12 jurors.
- Find the probability of 5 or fewer Mexican-Americans among 12 jurors.
- Which probability is relevant for determining whether 5 jurors among 12 is unusually low: the result from part (a) or part (b)?
- Is 5 an unusually low number of Mexican-Americans among 12 jurors? Why or why not?

**a.** P(5) = ** 0.003**

**b.** The probability of 8 or fewer Mexican-Americans
among 12 jurors is **0.004**.

*(0 + 0 + 0 + 0 + 0.001 + 0.003) = 0.004*

**c.** The result from part (b), because it measures the
probability of 8 or fewer successes.

**d.** Yes, because the relevant probability is less than
or equal to 0.05.

A Gallup poll of 1236 adults showed that 12% of the respondents believe that it is bad luck to walk under a ladder. Consider the probability that among 30 randomly selected people from the 1236 who were polled, there are at least 2 who have that belief. Given that the subjects surveyed were selected without replacement, the events are not independent.

Can the probability be found by using the binomial probability formula? Why or why not?

**Yes. Although the selections are not independent, they can be
treated as being independent by applying the 5% guideline.**

*Only 30 people are randomly selected from the group of 1236,
which is less than 5%.*

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.

Four 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.

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.

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

**a.** P(CWW) = **0.128**

*P(CWW) = P(C) x P(W) x P(W)*

*= (0.20) x (0.80) x (0.80) = 0.128*

**b.** P(WWC) = **0.128**
P(WCW) = **0.128**

*P(WWC) = P(W) x P(W) x P(C)*

*= 0.80 x 0.80 x 0.20 = 0.128*

*P(WCW) = P(W) x P(C) x P(W)*

*= 0.80 x 0.80 x 0.20 = 0.128*

**c. 0.384**

*P(CWW) + P(WWC) + P(WCW)*

*= 0.128 + 0.128 + 0.128*

*= 0.384*

Assume that a procedure yields a binomial distribution with 6 trials and a probability of success of 0.70.

Use a binomial probability table to find the probability that the number of successes is exactly 6.

**0.118**

*n = 6*

*p = 0.70*

*x = 6*

Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.

n = 20, x = 3, p = 0.25

P(3) = **0.134**

*P(x=3) = ( ^{n} _{x} ) x ( p )^{x} x ( q
)^{(n - x)}*

*= ( ^{ 20} _{3} ) x (0.25)^{3} x (0.75)^{17}*

*= 0.1338956152*

Refer to the accompanying technology display. The probabilities were obtained by entering the values of n equals 5 and p equals 0.716. In a clinical test of a drug, 71.6% of the subjects treated with 10 mg of the drug experienced headaches. In each case, assume that 5 subjects are randomly selected and treated with 10 mg of the drug.

Find the probability that at least four of the subjects experience headaches.

Is it unusual to have fewer than four subjects experience headaches?

The probability that at least four of the subjects experience
headaches is **0.5614**.

*P(x ≥ 4) = P(x=4 or x=5)*

*= 0.3732 + 0.1882*

*= 0.5614*

**No, because the probability that fewer than four subjects
will experience headaches is not unlikely.**

In a region, there is a 0.9 probability chance that a randomly selected person of the population has brown eyes. Assume 10 people are randomly selected.

- Find the probability that all of the selected people have brown eyes.
- Find the probability that exactly 9 of the selected people have brown eyes.
- Find the probability that the number of selected people that have brown eyes is 8 or more.
- If 10 people are randomly selected, is 8 an unusually high number for those with brown eyes?

**a.** The probability that all of the 10 selected people
have brown eyes is **0.349**.

*P(x = 10) = ( 0.9 ^{10 }) = 0.3486784401*

**b.** The probability that exactly 9 of the selected
people have brown eyes is ** 0.387**.

*P(x = 9) = ( ^{n} _{x} ) x ( p )^{x} x ( q
)^{(n - x)}*

*= ( ^{10} _{9}) x (0.9^{9}) x (0.1^{1})*

*= 0.387420489*

**c.** The probability that the number of selected people
that have brown eyes is 8 or more is ** 0.930**.

*P(x = 8) = ( ^{n} _{x} ) x ( p )^{x} x ( q
)^{(n - x)}*

*= ( ^{10} _{8}) x (0.9)^{8} x (0.1)^{2}*

*= 0.1937102445*

*P(x ≥ 8) = P(x=8 or x=9 or x=10)*

*= 0.194 + 0.387 + 0.349*

*= 0.930*

**d.** No, because the probability that 8 or more of the
selected people have brown eyes is greater than 0.05.

Researchers conducted a study to determine whether there were significant differences in graduation rates between medical students admitted through special programs and medical students admitted through the regular admissions criteria. It was found that the graduation rate was 91% for the medical students admitted through special programs in all medical schools.

- If 13 of the students admitted through special programs are randomly selected, find the probability that at least 12 of them graduated.
- Suppose a medical school has 13 students in one of the special programs of its medical program. Does the probability calculated in part (a) apply to these students?

**a.** The probability that at least 12 of the 13
students graduated is **0.670**.

*P(x ≥ 12) = P(x=12 or x=13)*

*= [ ( ^{13} _{12} ) x
(0.91)^{12} x (0.09)^{1} ] +
[ (0.91^{13}) ]*

*= 0.377 + 0.293*

*= 0.6704*

**b. **No, because the students admitted through a single
special program at a specific medical school are not a random sample.

When someone buys a ticket for an airline flight, there is a 0.0973 probability that the person will not show up for the flight. A certain jet can seat 17 passengers.

Is it wise to book 19 passengers for a flight on the jet? Explain.

It is **not a wise decision** because the probability
that there are not enough seats on the jet is ** 0.436**. So,
overbooking is **not an unlikely event**.

** n=19, q=0.0973, p=0.9027*

*P(x > 17) = P(x=18 or x=19)*

*= [ ( ^{19} _{18} ) x
(0.9027)^{18} x (0.0973)^{1} ] +
[ (0.9027)^{19} ]*

*= 0.293 + 0.143*

*= 0.4358*

The probability of a randomly selected adult in one country being infected with a certain virus is 0.003. In tests for the virus, blood samples from 26 people are combined.

What is the probability that the combined sample tests positive for the virus?

Is it unlikely for such a combined sample to test positive? Note that the combined sample tests positive if at least one person has the virus.

The probability that the combined sample will test positive is **0.075**.

*P(1 or 2 or ... or 26) = 1 – P(0)*

*= 1 – [ ( ^{26} _{} _{0}
) x (0^{}.003^{0}) x (0.997^{26}) ]*

*= 1 – 0.925*

*= 0.0751440049*

**It is not unlikely for such a combined sample to test
positive, because the probability that the combined sample will test
positive is greater than 0.05.**

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 23 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications.

If a particular shipment of thousands of aspirin tablets actually has a 3% rate of defects, what is the probability that this whole shipment will be accepted?

The probability that this whole shipment will be accepted is **0.849**.

*P(x=0 or x=1)*

*= [ ( ^{23} _{0} ) x
(0.03)^{0} x (0.97)^{23} ] +
[ ( ^{23} _{1} ) x
(0.03)^{1} x (0.97)^{22} ]*

*= 0.496 + 0.353*

*= 0.849*

A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 19 tablets. The entire shipment is accepted if at most 2 tablets do not meet the required specifications.

If a particular shipment of thousands of aspirin tablets actually has a 3.0% rate of defects, what is the probability that this whole shipment will be accepted?

The probability that this whole shipment will be accepted is **0.982**.

*P(x ≤ 2) = P(x=0 or x=1 or x=2)*

*= [ (0.97)^{19} ] +
[ ( ^{19} _{1} ) x
(0.03)^{1} x (0.97)^{18} ] +
[ (^{} ^{19} _{2} ) x
(0.03)^{2} x (0.97)^{17} ]*

*= 0.5606 + 0.3294 + 0.0917*

*= 0.9817*

Which of the following is not a requirement of the binomial probability distribution?

- The trials must be dependent.
- The procedure has a fixed number of trials.
- The probability of a success remains the same in all trials.
- Each trial must have all outcomes classified into two categories.

**The trials must be dependent.**

*"The trials must be dependent" is not a requirement of
a binomial probability distribution. For a binomial distribution,
the trials must be independent.*

Which of the following is NOT one of the three methods for finding binomial probabilities that is found in the chapter on discrete probability distributions?

- Use a simulation
- Use a statistical table for binomial probabilities
- Use the binomial probability formula
- Use computer software or a calculator

**Use a simulation**

*Using a simulation is not one of the suggested methods for finding
binomial probabilities.*

In a clinical trial of a cholesterol drug, 427 subjects were given a placebo, and 21% of them developed headaches. For such randomly selected groups of 427 subjects given a placebo, identify the values of n, p, and q that would be used for finding the mean and standard deviation for the number of subjects who develop headaches.

The value of n is **427**.

The value of p is ** 0.21**.

The value of q is **0.79**.

Assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p.

Use the given values of n and p to find the mean **μ**
and standard deviation **σ**.

Also, use the range rule of thumb to find the minimum usual value
**μ** – 2**σ** and the maximum usual value
**μ** + 2**σ**.

n = 1588, p = 3 / 4

**μ **= **1191**

*
μ = np*

*= (1588)(0.75)*

*= 1191*

**σ **= **17.3**

**σ** = √npq

= √(1588 x 0.75 x 0.25)

= 17.25543393

**μ** – 2**σ** = **1156.5**

**μ** + 2** σ **=

**1225.5**

Assume that the given procedure yields a binomial distribution with n trials and the probability of success for one trial is p.

Use the given values of n and p to find the mean **μ**
and standard deviation **σ**.

Also, use the range rule of thumb to find the minimum usual value
**μ** – 2**σ** and the maximum usual value
**μ** + 2**σ**.

In an analysis of preliminary test results from the a gender-selection method, 19 babies are born and it is assumed that 50% of babies are girls, so n = 19 and p = 0.5.

The value of the mean is **μ** = **9.5**.

* μ = np*

*= (19)(0.5)*

*= 9.5*

The value of the standard deviation **σ** = **2.2** .

**σ** = √npq

= √(19 x 0.5 x 0.5)

*= 2.179449472*

The minimum usual value is **μ** – 2**σ**
= **5.1**

The maximum usual value is **μ** + 2**σ**
= **13.9**

Several psychology students are unprepared for a surprise true/false test with 12 questions, and all of their answers are guesses.

- Find the mean and standard deviation for the number of correct answers for such students.
- Would it be unusual for a student to pass by guessing (which requires getting at least 10 correct answers)? Why or why not?

**a. μ** = **6**

*= (12)(0.5)*

*= 6*

**σ **= **1.7**

**σ** = √npq

= √(12 x 0.5 x 0.5)

= 1.732050808

**b.** Yes, because 10 is greater than the maximum usual value.

*
μ – 2σ = 2.6*

*
μ + 2σ = 9.4*

A headline in a certain newspaper states that "most stay at first job less than 2 years." That headline is based on an online poll of 390 college graduates. Among those polled, 76 % stayed at their first full-time job less than 2 years.

- Assuming that 50% is the true percentage of graduates who stay at their first job less than two years, find the mean and standard deviation of the numbers of such graduates in randomly selected groups of 390 graduates.
- Assuming that the 50% rate in part (a) is correct, find the range of usual values for the numbers of graduates among 390 who stay at their first job less than two years.
- Find the actual number of surveyed graduates who stayed at their first job less than two years. Use the range of values from part (b) to determine whether that number is unusual. Does the result suggest that the headline is not justified?
- The statement, "Alumni who opted-in to receive communications from the website were invited to participate in the online poll, and 390 of them completed the survey," was given as part of the description of the survey methods used. What does that statement suggest about the results?

**a. **The mean is
**195** graduates.

*
μ = np*

*= (390)(0.50)*

*= 195*

The standard deviation is
**9.9** graduates.

**σ** = √npq

= √(390 x 0.50 x 0.50)

= 9.874208829

**b.** The numbers of graduates among 390 who stay at
their first job less than two years should usually fall between **
175.2** and **
214.8**.

*
μ – 2σ = 175.2*

*
μ + 2σ = 214.8*

**c.** The actual number of surveyed graduates who stayed
at their first job less than two years is 296.

*390 x 76% = 296.4 (Round to the nearest whole number)*

Since the actual number is outside of the range of usual values found in part (b), it is unusual.

**The result suggests that the headline is justified.**

*Because the actual number of surveyed graduates who stayed at
their first job less than two years is larger than the maximum usual
value under the assumption that 50% is the true percentage.
Therefore, it is unlikely that the true percentage is 50% or less.*

**d.** Since the sample of 390 respondents is a
**voluntary response** sample, the results are **
very questionable**.

A government agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages. In standard English text, a particular letter is used at a rate of 6.8%.

- Find the mean and standard deviation for the number of times this letter will be found on a typical page of 1600 characters.
- In an intercepted message, a page of 1600 characters is found to have the letter occurring 116 times. Is this unusual?

**a.** **μ** =
**108.8**

*
μ = np*

*= (1600)(0.068)*

*= 108.8*

**σ** =
**10.1**

*
σ = √npq*

*= √(1600 x 0.068 x 0.932)*

*= 10.06983615*

**b.** No, because 116 is within the range of usual values.

*
μ – 2σ = 88.6*

*
μ + 2σ = 129.0*

If a gambler places a bet on the number 7 in roulette, he or she has a 1/38 probability of winning.

- Find the mean and standard deviation for the number of wins of gamblers who bet on the number 7 one hundred and seventy times.
- Would 0 wins in one hundred and seventy bets be an unusually low number of wins?

**a.** The value of the mean is **μ** =
**4.5**

*
μ = np*

*= (170)(1/38)*

*= 4.473684211*

The value of the standard deviation is **σ** =
**2.1**

*
σ = √npq*

*= √(170 x 1/38 x 37/38)*

*= 2.087092638*

**b.** Yes, because 0 is below the minimum usual value.

*
μ – 2σ = 0.3*

*
μ + 2σ = 8.7*

- For classes of 80 students, find the mean and standard deviation for the number born on the 4th of July. Ignore leap years.
- For a class of 80 students, would two be an unusually high number who were born on the 4th of July?

**a.** The value of the mean is **μ** =
**0.219178**

*
μ = np*

*= (80)(1/365)*

*= 0.2191780822*

The value of the standard deviation is **σ** =
**0.467523**

*
σ = √npq*

*= √(80 x 1/365 x 364/365)*

*= 0.4675228276*

**b.** Yes, because 2 is greater than the maximum usual value.

*
μ – 2σ = -0.715868 = 0*

*
μ + 2σ = 1.154224*

For the binomial distribution, which formula finds the standard deviation?

**
√npq**

*The formula √npq is the formula that finds the
standard deviation for the binomial distribution.*

Identify the expression for calculating the mean of a binomial distribution.

**np**

*The formula
μ = np is the expression that is equal to
the mean of a binomial probability distribution.*