Question 1

Find the indicated binomial probability. Round to five decimal places when necessary.
-In a certain college, 20% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that exactly 2 belong to an
Ethnic minority? 
A)1.8
B)0.30199
C)0.00671
D)0.00007

Accepted Answer

The probability can be calculated using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials (10 students), k is the number of successes (2 students belonging to an ethnic minority), and p is the probability of success on a single trial (20% or 0.20). Plugging in the values gives P(X = 2) = (10 choose 2) * (0.20)^2 * (0.80)^8, which calculates to approximately 0.30199.

Question 2

Solve the problem.
-A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a Poisson distribution with parameter $\lambda$ = 3.3. Construct a probability table for the random variable
X. Compute the probabilities for 0 - 5 sightings. 
A)
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.037 \
1 & 0.122 \
2 & 0.201 \
3 & 0.221 \
4 & 0.182 \
5 & 0.012
\end{array}$

B)
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.037 \
1 & 0.122 \
2 & 0.201 \
3 & 0.221 \
4 & 0.182 \
5 & 0.120
\end{array}$

C)
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.037 \
1 & 0.122 \
2 & 0.021 \
3 & 0.221 \
4 & 0.182 \
5 & 0.120
\end{array}$

D)
$\begin{array}{c|c}
\mathrm{x} & \mathrm{P}(\mathrm{X}=\mathrm{x}) \
\hline 0 & 0.370 \
1 & 0.122 \
2 & 0.201 \
3 & 0.221 \
4 & 0.182 \
5 & 0.120
\end{array}$

Accepted Answer

The probabilities for a Poisson distribution with parameter $\lambda = 3.3$ for 0 - 5 sightings are calculated using the formula $P(X=x) = \frac{e^{-\lambda} \lambda^x}{x!}$. The correct probabilities for 0 - 5 sightings match the values in option B.

Question 3

Find the standard deviation of the Poisson random variable. Round to three decimal places.
-Suppose X has a Poisson distribution with parameter $\lambda$ = 10.620. Find the standard deviation of X. 
A)10.620
B)5.310
C)3.259
D)112.784

Accepted Answer

The standard deviation of a Poisson random variable with parameter $\lambda$ is the square root of $\lambda$. Thus, $\sqrt{10.620} \approx 3.259$.

Question 4

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places.
-The probability that a car will have a flat tire while driving through a certain tunnel is 0.00005. Use the Poisson approximation to the binomial distribution to find the probability that among 8000 cars
Passing through this tunnel, at most two will have a flat tire. 
A)0.062
B)0.054
C)0.938
D)0.992

Accepted Answer

To use the Poisson approximation, first calculate the mean (λ) as the product of the number of trials (n) and the probability of success (p). Here, λ = 8000 * 0.00005 = 0.4. The probability of at most 2 cars having a flat tire is the sum of the probabilities of 0, 1, and 2 cars having a flat tire, calculated using the Poisson formula: P(X=k) = (e^(-λ) * λ^k) / k!, for k=0, 1, 2. Summing these probabilities gives a result close to 0.992.

Question 5

List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Accepted Answer

The answer of List the four requirements for a binomial...

Question 6

Find the indicated binomial probability. Round to five decimal places when necessary.
-A multiple choice test has 30 questions, and each has four possible answers, of which one is correct. If a student guesses on every question, find the probability of getting exactly 12 correct. 
A)0.00004
B)0.02906
C)5.1554
D)13,922,008.7

Accepted Answer

The probability of guessing exactly 12 correct answers out of 30 when each question has 4 options (with one correct answer) can be calculated using the binomial probability formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where n = 30, k = 12, p = 1/4 (since there's a 1 in 4 chance of guessing a question correctly), and C(n, k) is the combination of n items taken k at a time. Plugging these values into the formula gives a probability of approximately 0.02906.

Question 7

Find the indicated probability. Round to four decimal places.
-Find the probability of at least 2 girls in 6 births. Assume that male and female births are equally likely and that the births are independent events. 
A)0.6563
B)0.2344
C)0.8906
D)0.1094

Accepted Answer

The probability of at least 2 girls in 6 births can be found by subtracting the probability of getting 0 or 1 girl from 1. The probability of a girl in each birth is 0.5 (since male and female births are equally likely), and the events are independent. Using the binomial probability formula, P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the combination of n items taken k at a time, p is the probability of success (having a girl in this case), n is the total number of trials (births), and k is the number of successful outcomes (girls). The probability of 0 girls is C(6, 0) * (0.5)^0 * (0.5)^6 = (1) * (1) * (0.015625) = 0.015625. The probability of 1 girl is C(6, 1) * (0.5)^1 * (0.5)^5 = (6) * (0.5) * (0.03125) = 0.09375. Adding these probabilities gives 0.015625 + 0.09375 = 0.109375. Subtracting this sum from 1 gives the probability of having at least 2 girls: 1 - 0.109375 = 0.890625, which rounds to 0.8906.

Question 8

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary.
-$$\lambda = 3.1 ; \mathrm { P } ( \mathrm { X } \geq 2 )$$  
A)$0.599$
B) $0.185$
C) $0.216$
D) $0.815$

Accepted Answer

The probability that X is at least 2 is the complement of the probability that X is less than 2 (i.e., P(X=0) + P(X=1)). Using the Poisson formula, calculate P(X=0) and P(X=1), then subtract their sum from 1 to find P(X≥2). Calculation: P(X=0) = e^(-3.1)(3.1)^0/0! ≈ 0.045, P(X=1) = e^(-3.1)(3.1)^1/1! ≈ 0.140. Thus, P(X≥2) = 1 - (P(X=0) + P(X=1)) ≈ 1 - (0.045 + 0.140) = 0.815.

Question 9

What is the probability that 6 rolls of a fair die will show four exactly 2 times? 
A)0.0134
B)0.0067
C)0.41667
D)0.20094

Accepted Answer

The probability can be calculated using the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successful trials, p is the probability of success on a single trial, and (n choose k) is the binomial coefficient. For this problem, n = 6 (rolls), k = 2 (desired number of twos), and p = 1/6 (probability of rolling a two). Thus, P(X = 2) = (6 choose 2) * (1/6)^2 * (5/6)^4 = 15 * (1/36) * (625/1296) = 0.20094.

Question 10

Describe the Poisson distribution and give some example of a random variable with a Poisson distribution.

Accepted Answer

The answer of Describe the Poisson distribution and give some...

Question 11

A coin is biased. Danny wishes to determine the probability of obtaining heads when flipping this coin. He flips the coin 10 times and obtains 8 heads. He concludes that the probability of obtaining heads when flipping this coin is 0.8. Is his thinking reasonable? Why or why not?

Accepted Answer

The answer of A coin is biased. Danny wishes to...

Question 12

Find the indicated probability. Round to four decimal places.
-A car insurance company has determined that 9% of all drivers were involved in a car accident last year. Among the 10 drivers living on one particular street, 3 were involved in a car accident last
Year. If 10 drivers are randomly selected, what is the probability of getting 3 or more who were
Involved in a car accident last year? 
A)0.9548
B)0.0452
C)0.4435
D)0.0541

Accepted Answer

The probability of getting 3 or more drivers involved in an accident can be calculated using the binomial distribution with n=10, p=0.09. The probability of 0, 1, or 2 drivers involved is calculated and subtracted from 1 to find the probability of 3 or more.

Question 13

Find the indicated binomial probability. Round to five decimal places when necessary.
-A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely. 
A)0.16406
B)0.32813
C)0.00781
D)0.65625

Accepted Answer

The probability of exactly 5 females out of 7 kittens, assuming male and female births are equally likely, can be calculated using the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n=7, k=5, and p=0.5 (probability of a kitten being female). This calculation results in 0.16406.

Question 14

Determine the required probability by using the Poisson approximation to the binomial distribution. Round to threedecimal places.
-The probability that a car will have a flat tire while driving through a certain tunnel is 0.00004. Use the Poisson approximation to the binomial distribution to find the probability that among 11,000
Cars passing through this tunnel, at least one will have a flat tire. 
A)0.513
B)0.356
C)0.644
D)0.283

Accepted Answer

The answer of Determine the required probability by using the...

Question 15

The probability distribution of a random variable is given along with its mean and standard deviation. Draw aprobability histogram for the random variable; locate the mean and show one, two, and three standard deviationintervals.
-The random variable X is the number of tails when four coins are flipped. Its probability distribution is as follows. $\begin{array}{r|ccccc}
x & 0 & 1 & 2 & 3 & 4 \
\hline P(X=x) & \frac{1}{16} & \frac{1}{4} & \frac{3}{8} & \frac{1}{4} & \frac{1}{16}
\end{array}$
$$\mu = 2 , \sigma = 1$$

A)

B)

C)

Accepted Answer

The answer of The probability distribution of a random variable...

Question 16

Find the mean of the binomial random variable. Round to two decimal places when necessary.
-On a multiple choice test with 6 questions, each question has four possible answers, one of which is correct. For students who guess at all answers, find the mean for the random variable X, the
Number of correct answers. 
A)4.5
B)3
C)1.5
D)2

Accepted Answer

The mean of a binomial random variable is given by the formula μ = np, where n is the number of trials (in this case, 6 questions) and p is the probability of success on each trial (in this case, the probability of guessing a question correctly, which is 1/4 since there are four possible answers and only one is correct). Therefore, μ = 6 * (1/4) = 1.5.

Question 17

Find the standard deviation of the binomial random variable.
-According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size 
A)16. 2.75
B)3.52
C)1.66
D)1.88

Accepted Answer

The standard deviation of a binomial distribution is calculated using the formula: √(n * p * (1-p)), where n is the sample size and p is the probability of success. For n = 16 and p = 0.22, the standard deviation is √(16 * 0.22 * 0.78) ≈ 1.66.

Question 18

The random variable X represents the number of siblings of a student selected randomly from a particular college. Use random variable notation to express the following statement in shorthand. The probability that the student has two siblings is $0.18$.  
A)$\mathrm { P } ( \mathrm { X } = 2 ) = 0.18$
B) $P ( X ) = 0.18$
C) $\mathrm { P } ( 2 ) = 0.18$
D) $( X = 2 ) =$

Accepted Answer

A) correctly uses random variable notation to express the probability that the selected student has exactly two siblings as $0.18$.

Question 19

Provide an appropriate response.
-A die is rolled repeatedly until a six appears. The random variable X represents the total number of rolls preceding the six. What are the possible values of the random variable X? 
A)0, 1, 2, 3, 4, .....
B)0, 1, 2, 3, 4, 5, 6
C)1, 2, 3, 4, .....
D)1, 2, 3, 4, 5, 6

Accepted Answer

The random variable X represents the total number of rolls preceding the six. So, the possible values of X are 0, 1, 2, 3, 4, .....

Question 20

Construct a probability histogram for the binomial random variable, X.
-Two balls are drawn at random, with replacement, from a bag containing 4 red balls and 2 blue balls. X is the number of blue balls drawn. 
A)

B)

C)

D)

Accepted Answer

The answer of Construct a probability histogram for the binomial...

Quiz 5: Discrete Random Variables

Find the indicated binomial probability. Round to five decimal places when necessary. -In a certain college, 20% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that exactly 2 belong to an Ethnic minority?

Solve the problem. -A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a Poisson distribution with parameter $\lambda$ = 3.3. Construct a probability table for the random variable X. Compute the probabilities for 0 - 5 sightings.

Find the standard deviation of the Poisson random variable. Round to three decimal places. -Suppose X has a Poisson distribution with parameter $\lambda$ = 10.620. Find the standard deviation of X.

List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Find the indicated binomial probability. Round to five decimal places when necessary. -A multiple choice test has 30 questions, and each has four possible answers, of which one is correct. If a student guesses on every question, find the probability of getting exactly 12 correct.

Find the indicated probability. Round to four decimal places. -Find the probability of at least 2 girls in 6 births. Assume that male and female births are equally likely and that the births are independent events.

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - $\lambda = 3.1 ; \mathrm { P } ( \mathrm { X } \geq 2 )$

What is the probability that 6 rolls of a fair die will show four exactly 2 times?

Describe the Poisson distribution and give some example of a random variable with a Poisson distribution.

A coin is biased. Danny wishes to determine the probability of obtaining heads when flipping this coin. He flips the coin 10 times and obtains 8 heads. He concludes that the probability of obtaining heads when flipping this coin is 0.8. Is his thinking reasonable? Why or why not?

Find the indicated binomial probability. Round to five decimal places when necessary. -A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely.

Find the standard deviation of the binomial random variable. -According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size

The random variable X represents the number of siblings of a student selected randomly from a particular college. Use random variable notation to express the following statement in shorthand. The probability that the student has two siblings is $0.18$ .

Provide an appropriate response. -A die is rolled repeatedly until a six appears. The random variable X represents the total number of rolls preceding the six. What are the possible values of the random variable X?

Construct a probability histogram for the binomial random variable, X. -Two balls are drawn at random, with replacement, from a bag containing 4 red balls and 2 blue balls. X is the number of blue balls drawn.

Quiz 5: Discrete Random Variables

Find the indicated binomial probability. Round to five decimal places when necessary. -In a certain college, 20% of the physics majors belong to ethnic minorities. If 10 students are selected at random from the physics majors, what is the probability that exactly 2 belong to an Ethnic minority?

Solve the problem. -A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a Poisson distribution with parameter λ\lambdaλ = 3.3. Construct a probability table for the random variable X. Compute the probabilities for 0 - 5 sightings.

Find the standard deviation of the Poisson random variable. Round to three decimal places. -Suppose X has a Poisson distribution with parameter λ\lambdaλ = 10.620. Find the standard deviation of X.

List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements.

Find the indicated binomial probability. Round to five decimal places when necessary. -A multiple choice test has 30 questions, and each has four possible answers, of which one is correct. If a student guesses on every question, find the probability of getting exactly 12 correct.

Find the indicated probability. Round to four decimal places. -Find the probability of at least 2 girls in 6 births. Assume that male and female births are equally likely and that the births are independent events.

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - λ=3.1;P(X≥2) \lambda = 3.1 ; \mathrm { P } ( \mathrm { X } \geq 2 ) λ=3.1;P(X≥2)

What is the probability that 6 rolls of a fair die will show four exactly 2 times?

Describe the Poisson distribution and give some example of a random variable with a Poisson distribution.

A coin is biased. Danny wishes to determine the probability of obtaining heads when flipping this coin. He flips the coin 10 times and obtains 8 heads. He concludes that the probability of obtaining heads when flipping this coin is 0.8. Is his thinking reasonable? Why or why not?

Find the indicated binomial probability. Round to five decimal places when necessary. -A cat has a litter of 7 kittens. Find the probability that exactly 5 of the little furballs are female. Assume that male and female births are equally likely.

Find the standard deviation of the binomial random variable. -According to a college survey, 22% of all students work full time. Find the standard deviation for the random variable X, the number of students who work full time in samples of size

The random variable X represents the number of siblings of a student selected randomly from a particular college. Use random variable notation to express the following statement in shorthand. The probability that the student has two siblings is 0.180.180.18 .

Provide an appropriate response. -A die is rolled repeatedly until a six appears. The random variable X represents the total number of rolls preceding the six. What are the possible values of the random variable X?

Construct a probability histogram for the binomial random variable, X. -Two balls are drawn at random, with replacement, from a bag containing 4 red balls and 2 blue balls. X is the number of blue balls drawn.

Solve the problem. -A naturalist leads whale watch trips every morning in March. The number of whales seen X, has a Poisson distribution with parameter $\lambda$ = 3.3. Construct a probability table for the random variable X. Compute the probabilities for 0 - 5 sightings.

Find the standard deviation of the Poisson random variable. Round to three decimal places. -Suppose X has a Poisson distribution with parameter $\lambda$ = 10.620. Find the standard deviation of X.

Use the Poisson Distribution to find the indicated probability. Round to three decimal places when necessary. - $\lambda = 3.1 ; \mathrm { P } ( \mathrm { X } \geq 2 )$

The random variable X represents the number of siblings of a student selected randomly from a particular college. Use random variable notation to express the following statement in shorthand. The probability that the student has two siblings is $0.18$ .