Question 1

$$\text { Find the probability } P ( X ; \lambda ) : P ( 3,4 )$$

Accepted Answer

The probability $P(X; \lambda)$ for a Poisson distribution is given by $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. For $P(3,4)$, where $k=3$ and $\lambda=4$, the calculation is $P(3) = \frac{e^{-4} 4^3}{3!} = 0.1954$.

Question 2

The probability that a person will have 0, 1, or 2 dental checkups per year is 0.3 0.1, respectively. If seven people are picked at random, what is the probability that two
Will have no checkups, four will have one checkup, and one will have two checkups in
The next year?

Accepted Answer

The answer of The probability that a person will have...

Question 3

In a batch of 100 cell phones, there are, on average, 6 defective ones. If a random sample of 25 is selected, find the probability of 3 defective ones.

Accepted Answer

This problem can be solved using the hypergeometric distribution formula, which is appropriate for sampling without replacement from a finite population. The probability of getting exactly k defective items in a sample of size n from a population of size N with K defective items is given by the formula: P(X = k) = [C(K, k) * C(N-K, n-k)] / C(N, n), where C(n, k) is the combination of n items taken k at a time. Plugging in the values: P(3 defective) = [C(6, 3) * C(94, 22)] / C(100, 25), calculating this gives a probability of approximately 0.1303.

Question 4

In binomial experiments, each trial can be viewed as having only two outcomes.

Accepted Answer

The answer of In binomial experiments, each trial can be...

Question 5

A university has 10,000 students of which 35% are male and 65% are female. If a class of 30 students is chosen at random from the university population, find the mean and
Variance of the number of male students.

Accepted Answer

The answer of A university has 10,000 students of which...

Question 6

In the instructor's answer book for a mathematics text, 8% of the answers are incorrect. Use the Poisson approximation to express the probability that there are exactly 2
Incorrect answers for a homework set with 50 problems.

Accepted Answer

The Poisson approximation formula is $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the mean number of successes (in this case, incorrect answers) and $k$ is the number of successes we're interested in. Here, with 8% of 50 problems being incorrect on average, $\lambda = 0.08 \times 50 = 4$. For exactly 2 incorrect answers, $k = 2$, so the correct formula is $\frac{e^{-4} 4^2}{2!}$.

Question 7

In a binomial experiment, the outcome of each trial will depend on the ones before it.

Accepted Answer

The answer of In a binomial experiment, the outcome of...

Question 8

Use the multinomial formula to find the probability for the following situation.
$X _ { 1 } = 2 , X _ { 2 } = 2 , X _ { 3 } = 2$ when $n = 6 , p _ { 1 } = 0.3 , p _ { 2 } = 0.5 , p _ { 3 } = 0.2$

Accepted Answer

The multinomial formula is given by $\frac{n!}{x_1!x_2!...x_k!}p_1^{x_1}p_2^{x_2}...p_k^{x_k}$ for a multinomial distribution with n trials and probabilities $p_1, p_2, ..., p_k$ for the k outcomes. Here, with $n=6$, $x_1=x_2=x_3=2$, and $p_1=0.3$, $p_2=0.5$, $p_3=0.2$, the correct formula is $\frac{6!}{2!2!2!}(0.3)^2(0.5)^2(0.2)^2$.

Question 9

Use the multinomial formula to find the probability for the situation in which $n = 5 , X$ $= 2 , X _ { 2 } = 2 , X _ { 3 } = 1$, and $p _ { 1 } = 0.3 , p _ { 2 } = 0.4 , p _ { 3 } = 0.3$.

Accepted Answer

The multinomial probability is calculated using the formula: $\frac{n!}{x_1!x_2!x_3!} \cdot p_1^{x_1} \cdot p_2^{x_2} \cdot p_3^{x_3}$. Substituting the given values: $\frac{5!}{2!2!1!} \cdot 0.3^2 \cdot 0.4^2 \cdot 0.3^1 = 0.1296$.

Question 10

A binomial distribution is the probabilities of the possible outcomes of a binomial
experiment.

Accepted Answer

The answer of A binomial distribution is the probabilities of...

Question 11

Use the multinomial formula and find the probability for the following data.
$$\begin{array} { l } 
n = 8 , X _ { 1 } = 4 , X _ { 2 } = 3 , X _ { 3 } = 1 , \
p _ { 1 } = 0.30 , p _ { 2 } = 0.50 , p _ { 3 } = 0.20
\end{array}$$

Accepted Answer

The answer of Use the multinomial formula and find the...

Question 12

A computer store has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance
Of the number of ink jet printers.

Accepted Answer

The mean of the number of ink jet printers in a randomly chosen group of 10 from the store can be calculated using the formula for the expected value of a hypergeometric distribution, which is $n \times \frac{K}{N}$, where $n$ is the size of the group chosen (10 printers), $K$ is the total number of ink jet printers (50), and $N$ is the total number of printers (75). Thus, the mean is $10 \times \frac{50}{75} = 6.7$.The variance of the number of ink jet printers can be calculated using the formula for the variance of a hypergeometric distribution, which is $n \times \frac{K}{N} \times \left(1 - \frac{K}{N}\right) \times \frac{N-n}{N-1}$. Substituting the values, we get the variance as $10 \times \frac{50}{75} \times \left(1 - \frac{50}{75}\right) \times \frac{75-10}{75-1} = 2.2$.

Question 13

The probability that federal income tax returns will have 0, 1, or 2 errors is 0.73, 0.23, and 0.04, respectively. If 10 randomly selected returns are audited, what is the
Probability that eight will have no errors, two will have one error, and none will have
Two errors?

Accepted Answer

The answer of The probability that federal income tax returns...

Question 14

A school club consists of 26 male students and 11 female students. If 4 students are selected to represent the club in the student government, what is the probability
Male and 2 will be female?

Accepted Answer

The probability of selecting 2 male and 2 female students can be calculated using combinations. First, calculate the total number of ways to select 4 students out of 37, which is C(37,4). Then, calculate the number of ways to select 2 males out of 26 and 2 females out of 11, which is C(26,2) * C(11,2). The probability is the ratio of these two values: [C(26,2) * C(11,2)] / C(37,4) = 0.271.

Question 15

The failure rate for taking the bar exam in Philadelphia is 41%. If 375 people take the bar exam, what is the mean for the number of failures?

Accepted Answer

The mean number of failures can be calculated by multiplying the total number of people taking the exam (375) by the failure rate (41%), which gives 375 * 0.41 = 153.75, rounded to 153.8.

Question 16

The probability of a success remains the same for each trial in a binomial experiment.

Accepted Answer

The answer of The probability of a success remains the...

Question 17

The binomial distribution equations will not give exact probabilities for experiments in
which sampling is done without replacement. In those cases, the trials are not
independent.

Accepted Answer

The binomial distribution assumes that each trial is independent and the probability of success remains constant. When sampling is done without replacement, the probability of success changes with each trial, violating these assumptions.

Question 18

In a survey, 55% of the voters support a particular referendum. If 40 voters are
chosen at random, find the mean and variance for the number of voters who
support the referendum.

Accepted Answer

The answer of In a survey, 55% of the voters...

Question 19

A school is sending 12 children to a camp. If 20% of the children in the school
are first graders, and the 12 children are selected at random from among all 6
grades at the school, find the mean and variance of the number of first graders
chosen?

Accepted Answer

The answer of A school is sending 12 children to...

Question 20

Use the multinomial formula and find the probability for the following data.
$$\begin{array} { l } 
n = 6 , X _ { 1 } = 3 , X _ { 2 } = 2 , X _ { 3 } = 1 , \
p _ { 1 } = 0.58 , p _ { 2 } = 0.25 , p _ { 3 } = 0.17
\end{array}$$

Accepted Answer

The multinomial probability formula is given by:$$ P(X_1 = x_1, X_2 = x_2, ..., X_k = x_k) = \frac{n!}{x_1!x_2!...x_k!}p_1^{x_1}p_2^{x_2}...p_k^{x_k} $$Given $n = 6, X_1 = 3, X_2 = 2, X_3 = 1, p_1 = 0.58, p_2 = 0.25, p_3 = 0.17$, we substitute these values into the formula:$$ P = \frac{6!}{3!2!1!}(0.58)^3(0.25)^2(0.17)^1 $$$$ P = \frac{720}{12 \times 2 \times 1}(0.195112)(0.0625)(0.17) $$$$ P = 30 \times 0.195112 \times 0.0625 \times 0.17 $$$$ P \approx 0.124 $$Therefore, the correct answer is 0.124.

Quiz 5: Discrete Probability Distributions

$\text { Find the probability } P ( X ; \lambda ) : P ( 3,4 )$

The probability that a person will have 0, 1, or 2 dental checkups per year is 0.3 0.1, respectively. If seven people are picked at random, what is the probability that two Will have no checkups, four will have one checkup, and one will have two checkups in The next year?

In a batch of 100 cell phones, there are, on average, 6 defective ones. If a random sample of 25 is selected, find the probability of 3 defective ones.

In binomial experiments, each trial can be viewed as having only two outcomes.

A university has 10,000 students of which 35% are male and 65% are female. If a class of 30 students is chosen at random from the university population, find the mean and Variance of the number of male students.

In the instructor's answer book for a mathematics text, 8% of the answers are incorrect. Use the Poisson approximation to express the probability that there are exactly 2 Incorrect answers for a homework set with 50 problems.

In a binomial experiment, the outcome of each trial will depend on the ones before it.

Use the multinomial formula to find the probability for the following situation. $X _ { 1 } = 2 , X _ { 2 } = 2 , X _ { 3 } = 2$ when $n = 6 , p _ { 1 } = 0.3 , p _ { 2 } = 0.5 , p _ { 3 } = 0.2$

Use the multinomial formula to find the probability for the situation in which $n = 5 , X$ $= 2 , X _ { 2 } = 2 , X _ { 3 } = 1$ , and $p _ { 1 } = 0.3 , p _ { 2 } = 0.4 , p _ { 3 } = 0.3$ .

A binomial distribution is the probabilities of the possible outcomes of a binomial experiment.

Use the multinomial formula and find the probability for the following data. $\begin{array} { l } n = 8 , X _ { 1 } = 4 , X _ { 2 } = 3 , X _ { 3 } = 1 , \\p _ { 1 } = 0.30 , p _ { 2 } = 0.50 , p _ { 3 } = 0.20\end{array}$

A computer store has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance Of the number of ink jet printers.

The probability that federal income tax returns will have 0, 1, or 2 errors is 0.73, 0.23, and 0.04, respectively. If 10 randomly selected returns are audited, what is the Probability that eight will have no errors, two will have one error, and none will have Two errors?

A school club consists of 26 male students and 11 female students. If 4 students are selected to represent the club in the student government, what is the probability Male and 2 will be female?

The failure rate for taking the bar exam in Philadelphia is 41%. If 375 people take the bar exam, what is the mean for the number of failures?

The probability of a success remains the same for each trial in a binomial experiment.

The binomial distribution equations will not give exact probabilities for experiments in which sampling is done without replacement. In those cases, the trials are not independent.

In a survey, 55% of the voters support a particular referendum. If 40 voters are chosen at random, find the mean and variance for the number of voters who support the referendum.

A school is sending 12 children to a camp. If 20% of the children in the school are first graders, and the 12 children are selected at random from among all 6 grades at the school, find the mean and variance of the number of first graders chosen?

Use the multinomial formula and find the probability for the following data. $\begin{array} { l } n = 6 , X _ { 1 } = 3 , X _ { 2 } = 2 , X _ { 3 } = 1 , \\p _ { 1 } = 0.58 , p _ { 2 } = 0.25 , p _ { 3 } = 0.17\end{array}$

Quiz 5: Discrete Probability Distributions

Find the probability P(X;λ) :P(3,4) \text { Find the probability } P ( X ; \lambda ) : P ( 3,4 ) Find the probability P(X;λ) :P(3,4)

The probability that a person will have 0, 1, or 2 dental checkups per year is 0.3 0.1, respectively. If seven people are picked at random, what is the probability that two Will have no checkups, four will have one checkup, and one will have two checkups in The next year?

In a batch of 100 cell phones, there are, on average, 6 defective ones. If a random sample of 25 is selected, find the probability of 3 defective ones.

In binomial experiments, each trial can be viewed as having only two outcomes.

A university has 10,000 students of which 35% are male and 65% are female. If a class of 30 students is chosen at random from the university population, find the mean and Variance of the number of male students.

In the instructor's answer book for a mathematics text, 8% of the answers are incorrect. Use the Poisson approximation to express the probability that there are exactly 2 Incorrect answers for a homework set with 50 problems.

In a binomial experiment, the outcome of each trial will depend on the ones before it.

Use the multinomial formula to find the probability for the following situation. X1=2,X2=2,X3=2X _ { 1 } = 2 , X _ { 2 } = 2 , X _ { 3 } = 2X1​=2,X2​=2,X3​=2 when n=6,p1=0.3,p2=0.5,p3=0.2n = 6 , p _ { 1 } = 0.3 , p _ { 2 } = 0.5 , p _ { 3 } = 0.2n=6,p1​=0.3,p2​=0.5,p3​=0.2

Use the multinomial formula to find the probability for the situation in which n=5,Xn = 5 , Xn=5,X =2,X2=2,X3=1= 2 , X _ { 2 } = 2 , X _ { 3 } = 1=2,X2​=2,X3​=1 , and p1=0.3,p2=0.4,p3=0.3p _ { 1 } = 0.3 , p _ { 2 } = 0.4 , p _ { 3 } = 0.3p1​=0.3,p2​=0.4,p3​=0.3 .

A binomial distribution is the probabilities of the possible outcomes of a binomial experiment.

A computer store has 75 printers of which 25 are laser printers and 50 are ink jet printers. If a group of 10 printers is chosen at random from the store, find the mean and variance Of the number of ink jet printers.

The probability that federal income tax returns will have 0, 1, or 2 errors is 0.73, 0.23, and 0.04, respectively. If 10 randomly selected returns are audited, what is the Probability that eight will have no errors, two will have one error, and none will have Two errors?

A school club consists of 26 male students and 11 female students. If 4 students are selected to represent the club in the student government, what is the probability Male and 2 will be female?

The failure rate for taking the bar exam in Philadelphia is 41%. If 375 people take the bar exam, what is the mean for the number of failures?

The probability of a success remains the same for each trial in a binomial experiment.

The binomial distribution equations will not give exact probabilities for experiments in which sampling is done without replacement. In those cases, the trials are not independent.

In a survey, 55% of the voters support a particular referendum. If 40 voters are chosen at random, find the mean and variance for the number of voters who support the referendum.

A school is sending 12 children to a camp. If 20% of the children in the school are first graders, and the 12 children are selected at random from among all 6 grades at the school, find the mean and variance of the number of first graders chosen?

$\text { Find the probability } P ( X ; \lambda ) : P ( 3,4 )$

Use the multinomial formula to find the probability for the following situation. $X _ { 1 } = 2 , X _ { 2 } = 2 , X _ { 3 } = 2$ when $n = 6 , p _ { 1 } = 0.3 , p _ { 2 } = 0.5 , p _ { 3 } = 0.2$

Use the multinomial formula to find the probability for the situation in which $n = 5 , X$ $= 2 , X _ { 2 } = 2 , X _ { 3 } = 1$ , and $p _ { 1 } = 0.3 , p _ { 2 } = 0.4 , p _ { 3 } = 0.3$ .