Find the probability for the experiment of drawing two marbles (without replacement) at random from a bag containing one green, two yellow, and three red marbles.
1. Both marbles are red.
2. Both marbles are yellow.
3. Neither marble is yellow.
4. The marbles are of different colors.

The two vectors which are orthogonal to the given vector u are (0, 4, 1) and  (8, 0, -5).

Use the concept of orthogonal vectors defined as:

The term "orthogonal" in mathematics denotes a direction at a 90° angle. If two vectors u, and v form a right angle when they are perpendicular, or if the dot product they produce is zero, then they are said to be orthogonal.

The given vector is,

u = (5, −2, 8)

Let the vector orthogonal to this is,

v = (x, y, z)

Since vectors u and v are orthogonal,

Therefore,

u.v = 0

(5, −2, 8)(x, y, z) = 0

5x -2y + 8x = 0  .....(i)

To find the vectors choose x, y, and x such that equation one is satisfied.

So take x = 0, y = 4 and z = 1

Then (x, y, z) = (0, 4, 1) satisfy the equation (i)

Similarly, choose x = 8, y = 0, z = -5

Then (x, y, z) = (8, 0, -5) satisfy the equation (i)

Hence,

Two orthogonal vectors are (0, 4, 1) and  (8, 0, -5)

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Two vectors orthogonal to u=(5, -2, 8) could be v=(2, 5, 0) and u'=(-2, -5, 0). Both vectors satisfy the definition of orthogonality, having a dot product with u equal to zero.

To find two vectors that are orthogonal to the given vector u = (5, -2, 8), we need to find two vectors that have a dot product with vector u equal to zero. This is because orthogonality (perpendicularity in 3D space) is defined by a zero dot product.

For instance, let's calculate the dot product of u with v = (2, 5, 0). It'll be (5*2) + ((-2)*5) + (8*0) = 0. Therefore, vector v = (2, 5, 0) is orthogonal to u.

For a vector in the opposite direction, we simply need to multiply every component of vector v by -1. Consequently, a vector u′ = (-2, -5, 0) is in the opposite direction of v but still orthogonal to u. So, our answer is vectors v = (2, 5, 0) and u' = (-2, -5, 0).

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Answer:

a) 16% of students have an SAT math score greater than 615.

b) 2.5% of students have an SAT math score greater than 715.

c) 34% of students have an SAT math score between 415 and 515.

d) [tex]Z = 1.05[/tex]

e) [tex]Z = -1.10[/tex]

Step-by-step explanation:

To solve this question, we have to understand the normal probability distribution and the empirical rule.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Empirical rule

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

[tex]\mu = 515, \sigma = 100[/tex]

(a) What percentage of students have an SAT math score greater than 615?

615 is one standard deviation above the mean.

68% of the measures are within 1 standard deviation of the mean. The other 32% are more than 1 standard deviation from the mean. The normal probability distribution is symmetric. So of those 32%, 16% are more than 1 standard deviation above the mean and 16% more then 1 standard deviation below the mean.

So, 16% of students have an SAT math score greater than 615.

(b) What percentage of students have an SAT math score greater than 715?

715 is two standard deviations above the mean.

95% of the measures are within 2 standard deviations of the mean. The other 5% are more than 2 standard deviations from the mean. The normal probability distribution is symmetric. So of those 5%, 2.5% are more than 2 standard deviations above the mean and 2.5% more then 2 standard deviations below the mean.

So, 2.5% of students have an SAT math score greater than 715.

(c) What percentage of students have an SAT math score between 415 and 515?

415 is one standard deviation below the mean.

515 is the mean

68% of the measures are within 1 standard deviation of the mean. The normal probability distribution is symmetric, which means that of these 68%, 34% are within 1 standard deviation below the mean and the mean, and 34% are within the mean and 1 standard deviation above the mean.

So, 34% of students have an SAT math score between 415 and 515.

(d) What is the z-score for student with an SAT math score of 620?

We have that:

[tex]\mu = 515, \sigma = 100[/tex]

This is Z when X = 620. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{620 - 515}{100}[/tex]

[tex]Z = 1.05[/tex]

(e) What is the z-score for a student with an SAT math score of 405?

We have that:

[tex]\mu = 515, \sigma = 100[/tex]

This is Z when X = 405. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{405 - 515}{100}[/tex]

[tex]Z = -1.10[/tex]

a. Approximately 15.87% of students have an SAT math score greater than 615.

b. Approximately 2.28% of students have an SAT math score greater than 715.

c. 68% percentage of students have an SAT math score between 415 and 515

d. 1.05 is the z-score for student with an SAT math score of 620

e. -1.1 is the z-score for a student with an SAT math score of 405

To answer these questions, we can use the properties of a bell-shaped distribution and the empirical rule. The empirical rule states that for a bell-shaped distribution:Approximately 68% of the data falls within one standard deviation of the mean. Approximately 95% of the data falls within two standard deviations of the mean.Approximately 99.7% of the data falls within three standard deviations of the mean.

Given information: Mean (μ) = 515 and Standard Deviation (σ) = 100

(a) To find this, we need to calculate the z-score for a score of 615 and then find the percentage of data above that z-score using the standard normal distribution table.

z-score = (X - μ) / σ

z-score = (615 - 515) / 100

z-score = 1

Using the standard normal distribution table (or calculator), we find that approximately 84.13% of the data is below a z-score of 1. Since the distribution is symmetric, the percentage above the z-score of 1 is also approximately 100% - 84.13% = 15.87%.

Therefore, approximately 15.87% of students have an SAT math score greater than 615.

(b) We repeat the same process for a score of 715.

z-score = (715 - 515) / 100

z-score = 2

Using the standard normal distribution table (or calculator), we find that approximately 97.72% of the data is below a z-score of 2. The percentage above the z-score of 2 is approximately 100% - 97.72% = 2.28%.

Therefore, approximately 2.28% of students have an SAT math score greater than 715.

(c) We can use the empirical rule to find the percentage of students within one standard deviation of the mean and then subtract that from 100% to find the percentage between 415 and 515.

Percentage between 415 and 515 ≈ 68%

(d) We can calculate the z-score as follows:

z-score = (620 - 515) / 100

z-score = 1.05

(e) We can calculate the z-score as follows:

z-score = (405 - 515) / 100

z-score = -1.1

Remember that a negative z-score indicates a value below the mean.

Note: These calculations assume a normal distribution, and the actual percentages may vary slightly due to the discrete nature of test scores and rounding in calculations.

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Answer: for 10th percentile, X = 60.53inches

for 80th percentile, X = 61.53 inches

Step-by-step explanation:

the relationship between the mean, standard deviation and the standard normal distribution is given as

X = μ + σZ

where μ is the mean and σ is the standard deviation of the variable X, and Z is the value from the standard normal distribution for the desired percentile.

Hence to determine the 10th and 80th percentile, we lookup the standard normal distribution table attached below,

from the table,

at 10th percentile Z = -1.282

at 80th percentile we interpolate between 75th percentile and 90th

(80 - 75)/(90 - 75) = (Z - 0.675)/(1.282 - 0.675)

5/15 = Z - 0.675/0.607

0.333*(0.607) = Z - 0.675

Z = 0.8771

hence the Z value for the 80th percentile is 0.8771

hence

X value for 10th percentile and 80th is calculated as

X = μ + σZ

since mean = 63.7 and standard deviation = 2.47

For 10th percentile

X = 63.7 + 2.47*(-1.282)

X = 60.53

for 80th percentile

X = 63.7 + 2.47*(0.8771)

X = 61.53

Final answer:

Height at 10th percentile: 61.53 inches

Height at 80th percentile: 65.78 inches

Explanation:

Given that the heights of women are normally distributed with a mean of 63.7 inches and a standard deviation of 2.47 inches, we can find the 10th and 80th percentiles using the Z-score formula in the context of a normal distribution.

The Z-score formula is given by: Z = (X - μ) / σ, where X is the value whose Z-score we're calculating, μ is the mean, and σ is the standard deviation.

To find the 10th and 80th percentiles, we first use Z-scores corresponding to these percentiles from a standard normal distribution table:

We then apply the formula for each Z-score to find the heights corresponding to these percentiles:

Answer:

(a) The histogram is shown below.

(b) E (X) = 4.2

(c) SD (X) = 2.73

Step-by-step explanation:

Let X = r = a driver will tailgate the car in front of him before passing.

The probability that a driver will tailgate the car in front of him before passing is, P (X) = p = 0.35.

The sample selected is of size n = 12.

The random variable X follows a Binomial distribution with parameters n = 12 and p = 0.35.

The probability function of a binomial random variable is:

[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x}[/tex]

(a)

For X = 0 the probability is:

[tex]P(X=0)={12\choose 0}(0.35)^{0}(1-0.35)^{12-0}=0.006[/tex]

For X = 1 the probability is:

[tex]P(X=1)={12\choose 1}(0.35)^{1}(1-0.35)^{12-1}=0.037[/tex]

For X = 2 the probability is:

[tex]P(X=2)={12\choose 2}(0.35)^{2}(1-0.35)^{12-2}=0.109[/tex]

Similarly the remaining probabilities will be computed.

The probability distribution table is shown below.

The histogram is also shown below.

(b)

The expected value of a Binomial distribution is:

[tex]E(X)=np[/tex]

The expected number of vehicles out of 12 that will tailgate is:

[tex]E(X)=np=12\times0.35=4.2[/tex]

Thus, the expected number of vehicles out of 12 that will tailgate is 4.2.

(c)

The standard deviation of a Binomial distribution is:

[tex]SD(X)=np(1-p)[/tex]

The standard deviation of vehicles out of 12 that will tailgate is:

[tex]SD(X)=np(1-p)=12\times0.35\times(1-0.35)=2.73\\[/tex]

Thus, the standard deviation of vehicles out of 12 that will tailgate is 2.73.

To determine the probability distribution, create a histogram showing the possible values of r and their probabilities. The expected number of vehicles that will tailgate can be calculated by multiplying each value of r by its probability and summing up the results. The standard deviation can be found by calculating the variance and taking the square root of it.

In order to determine the probability distribution of the number of vehicles that tailgate before passing, we need to consider the given information. We know that about 35% of all drivers tailgate. Since we are passed by 12 vehicles, the number of vehicles that tailgate can range from 0 to 12. To create a histogram showing the probability distribution, we need to calculate the probability of each possible value of r and represent them in a bar graph.

(b) To compute the expected number of vehicles that will tailgate, we need to multiply each possible value of r by its corresponding probability and sum up the results. This will give us the average number of vehicles that tailgate out of the 12 vehicles that passed us.

(c) The standard deviation of this distribution can be calculated by determining the variance and taking the square root of it. Variance is calculated by summing up the squared differences between each value of r and the expected value, multiplying each squared difference by its corresponding probability, and summing up the results.

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Answer:

Correct answer is (c). 5

Step-by-step explanation:

It is important to note that solving problem requires techniques and intelligent people most especially when problem are complex or hard in nature. It is therefore important to ensure the numbers of problem solving experts should not be undersized than required to avoid over burden them and should not be too large to avoid conflict in their individual resolutions. Hence, most scientific reports state that problem solving experts should be within 3 to 5 members and as for this question, the optimum is 5 members.

Full Question

The number of messages that arrive at a Web site is a Poisson distributed random variable with a mean of 6 messages per hour.

a. What is the probability that 6 messages are received in 1 hour?

b. What is the probability that 10 messages are received in 1.5 hours?

c. What is the probability that fewer than 2 messages are received in 0.5 hour?

Answer and Explanation

Given

λ = 6 per hour

Poisson Probability P(X = k) = (λ^k e^-λ)/k!

a. K = 6

P(X = 6) = (6^6 e^-6)/6!

P(X = 6) = 0.160623141047980

P(X = 6) = 0.1606--------- Approximated

b.

If 6 messages are received on average per hour then the number of messages received on average per 1.5 hours is

λ = 6 *1.5

λ = 9

For k = 10

P(X = 10) = (9^10 e^-9)/10!

P(X = 10) = 0.118580076008570

P(X=10) = 0.1186 ---------- Approximated

c.

If 6 messages are received on average per hour then the number of messages received on average per 0.5 hours is

λ = 6 *0.5

λ = 3

For messages fewer than 2 means than k = 0 or k = 1

For k = 0

P(X = 0) = (3^0 e^-3)/0!

P(X = 0) = 0.049787068367863

P(X = 0) = 0.0498 ------_--- Approximated

For X = 1

P(X = 1) = (3^1 e^-3)/1!

P(X = 1) = 0.149361205103591

P(X = 1) = 0.1494 ---------- Approximated

P(X <2) = P(X=0) + P(X=1)

P(X<2) = 0.0498 + 0.1494

P(X<2) = 0.1996

Answer:

(a) 0.01625

(b) 0.3483

(c)  0.15170

Step-by-step explanation:

Given that there are total of 20 study problems from which 10 will come in the exam and A student knows how to solve 15 of the problems from those 20 total problems.

(a) To calculate the probability that the student will be able to answer all 10 questions on the exam, we know that:

students will be able answer all 10 questions in the exam only when all these 10 questions will be from the 15 problem which he know how to solve.

So the chances that he knows all 10 questions in the exam = [tex]^{15}C_1_0[/tex]

And total ways in which he answer 10 question from the 20 study problems =[tex]^{20}C_1_0[/tex]

Therefore, the Probability that the student will be able to answer all 10 questions on the exam = [tex]\frac{^{15}C_1_0}{^{20}C_1_0}[/tex] = [tex]\frac{15!}{5!\times 10!} \times \frac{10!\times 10!}{20!}[/tex] {Because [tex]^{n}C_r[/tex] = [tex]\frac{n!}{r!\times (n-r)!}[/tex] }

                                                  = 0.01625

(b) Probability that the student will be able to answer exactly eight questions on the exam = In the numerator there will be No. of ways that he answer exactly eight questions from the 15 problems he knows and the remaining 2 questions he solve from the 5 questions whose answer he doesn't know and in the denominator there will Total number of ways in which he answer 10 question from the 20 study problems.

So Required Probability = [tex]\frac{^{15}C_8 \times ^{5}C_2 }{^{20}C_1_0 }[/tex]  = [tex]\frac{15!}{8!\times 7!}\times \frac{5!}{2!\times 3!}\times \frac{10!\times 10!}{20!}[/tex] = 0.3483

(c) To calculate the probability that student will be able to answer at least nine questions on the exam is given by that [He will be able to nine questions on the exam + He will be able to answer all  10 questions on the exam]

So no. of ways that he will be able to nine questions on the exam = He answer 9 questions from those 15 problems which he know and remaining one question from the other 5 questions we he don't know = [tex]^{15}C_9\times ^{5}C_1[/tex]

And no. of ways that he will be able to answer all  10 questions on the exam

= [tex]^{15}C_1_0[/tex]  

So, the required probability = [tex]\frac{(^{15}C_9\times ^{5}C_1)+^{15}C_1_0}{^{20}C_1_0}[/tex] = 0.15170

The probability that the student will be able to answer all 10 questions on the exam is 0.01625.

The probability that the student will be able to answer all 10 questions on the exam will be:

= 15C10 / 20C10

= [15! / (5! × 10!)] × [10! × 20!/20!]

= 0.01625

The probability that the student will be able to answer exactly eight questions on the exam will be:

= (15C8 × 5C2) / 20C10

= 0.3483

The probability that the student will answer at least nine questions on the exam will be:

= [(15C9 × 5C1) + 15C10] / 20C10]

= 0.15170

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Answer:

[tex]P(X<480)=P(\frac{X-\mu}{\sigma}<\frac{480-\mu}{\sigma})=P(Z<\frac{480-412}{68})=P(z<1)[/tex]

[tex]P(z<1)=0.841[/tex]

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(412,68)[/tex]  

Where [tex]\mu=412[/tex] and [tex]\sigma=68[/tex]

We are interested on this probability

[tex]P(X<480)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X<480)=P(\frac{X-\mu}{\sigma}<\frac{480-\mu}{\sigma})=P(Z<\frac{480-412}{68})=P(z<1)[/tex]

And we can find this probability using the normal standard table or excel and we got:

[tex]P(z<1)=0.841[/tex]

Answer: The markup on the T-shirts is $ 13.50.

Step-by-step explanation:

Given : The cost price of each t-shirt = $1.50

The selling price of each t-shirt = $15

Then ,the markup on the T-shirts =  (Selling price of each t-shirt) -( Cost price of each t-shirt)

i.e. The markup on the T-shirts = $15- $1.50= $ 13.50

Hence, the markup on the T-shirts is $ 13.50.

Answer:

90%

Step-by-step explanation:

(a) The inverse of 1234 (mod 4321) is x such that 1234*x ≡ 1 (mod 4321). Apply Euclid's algorithm:

4321 = 1234 * 3 + 619

1234 = 619 * 1 + 615

619 = 615 * 1 + 4

615 = 4 * 153 + 3

4 = 3 * 1 + 1

Now write 1 as a linear combination of 4321 and 1234:

1 = 4 - 3

1 = 4 - (615 - 4 * 153) = 4 * 154 - 615

1 = 619 * 154 - 155 * (1234 - 619) = 619 * 309 - 155 * 1234

1 = (4321 - 1234 * 3) * 309 - 155 * 1234 = 4321 * 309 - 1082 * 1234

Reducing this leaves us with

1 ≡ -1082 * 1234 (mod 4321)

and so the inverse is

-1082 ≡ 3239 (mod 4321)

(b) Both 24140 and 40902 are even, so there GCD can't possibly be 1 and there is no inverse.

The multiplicative inverse of a number is simply its reciprocal

To determine the multiplicative inverse of a mod b, one of a and b must not be an even number

(a) Multiplicative inverse of 1234 mod 4321

This can be written as:

[tex]\mathbf{1234 \times x \equiv 1\ (mod\ 4321)}[/tex]

When the extended Euclidean's algorithm is applied, we start by writing the expression in the following format:

[tex]\mathbf{Dividend = Quotient \times Divisor + Remainder}[/tex]

So, we have:

[tex]\mathbf{4321 = 1234 \times 3 + 619}[/tex]

Express 1234 using the above format

[tex]\mathbf{1234 = 619 \times 1 + 615}[/tex]

Repeat the process for all quotient

[tex]\mathbf{619 = 615 \times 1 + 4}[/tex]

[tex]\mathbf{615 = 4 \times 153 + 3}[/tex]

[tex]\mathbf{4= 3 \times 1 + 1}[/tex]

Next, we reverse the process as follows:

Make 1 the subject in [tex]\mathbf{4= 3 \times 1 + 1}[/tex]

[tex]\mathbf{1 = 4 - 3}[/tex]

Substitute an equivalent expression for 3

[tex]\mathbf{1 = 4 - (615 - 4 \times 153)}[/tex]

[tex]\mathbf{1 = 4 - 615 + 4 \times 153}[/tex]

Collect like terms

[tex]\mathbf{1 = 4 + 4 \times 153 - 615 }[/tex]

[tex]\mathbf{1 = 4 \times 154 - 615 }[/tex]

Substitute an equivalent expression for 615

[tex]\mathbf{1 = 619 \times 154 - 155 \times (1234 - 619) }[/tex]

[tex]\mathbf{1 = 619 \times 309 - 155 \times 1234 }[/tex]

Substitute an equivalent expression for 619

[tex]\mathbf{1 =(4321 - 1234 \times 3) \times 309 - 155 \times 1234}[/tex]

[tex]\mathbf{1 = 4321 \times 309 - 1082 \times 1234}[/tex]

Recall that:

[tex]\mathbf{1234 \times x \equiv 1\ (mod\ 4321)}[/tex]

So, we have:

[tex]\mathbf{1 \equiv -1082 \times 1234\ mod(4321)}[/tex]

Add 4321 and -1082

[tex]\mathbf{4321 -1082 = 3239}[/tex]

Hence, the required inverse is:

[tex]\mathbf{ -1082 \equiv 3239\ (mod\ 4321)}[/tex]

(b) Multiplicative inverse of 24140 mod 40902

Recall that:

To determine the multiplicative inverse of a mod b, one of a and b must not be an even number

Because 24140 and 40902 are both even numbers, then:

24140 mod 40902 has no multiplicative inverse

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Answer:

1300 millimeters

Step-by-step explanation:

Answer:

1300 mm

Step-by-step explanation:

The probability that a device using five silicon chips selected randomly from a batch of 100 chips, which includes five defective ones, contains no defective chip is calculated by the ratio of selecting five good chips to selecting any five chips from the batch.

The question is asking to find the probability that a device, which uses five silicon chips selected from a batch of a hundred chips with five being defective, will have no defective chip. To solve this, we can calculate the probability step by step using the concept of combinations.

Firstly, we determine the number of ways to select five non-defective chips out of 95 good ones, which is C(95,5). Then, we calculate the total number of ways to select any five chips out of the whole batch, which is C(100,5). The probability that the device contains no defective chip is the ratio of these two numbers:

P(device has no defective chip) = C(95,5) / C(100,5)

Where C(n,k) represents the number of combinations of n items taken k at a time.

To calculate this, use factorials where C(n,k) = n! / [(n-k)!k!].

So, the probability that the device contains no defective chip, is:

P(device has no defective chip) = (95! / (90!*5!)) / (100! / (95!*5!))

Simplifying the factorials, we have:

P(device has no defective chip) = (95*94*93*92*91) / (100*99*98*97*96)

Finally, calculate this to get the decimal value, which would give the probability that the device contains no defective chips when made up from one batch.

Answer:

The given value is discrete  variable.

Step-by-step explanation:

Discrete Variable:

Discrete Variable are those variables that can only take on a finite number of values are called "discrete variables." All qualitative variables are discrete. Some quantitative variables are discrete, such as performance rated as 1,2,3,4, or 5, or temperature rounded to the nearest degree.

Here They have visited the doctor many times so it will be a whole number for sure.

Answer:

a) E(X) = -$0.0813 , s.d (X) = 3

b) E(X) = -$0.0813 , s.d (X) = 3

c) expected loss and higher stakes of loosing.

Step-by-step explanation:

Given:

- There are total 37 slots:

           Red = 18

           Black = 18

           Green = 1

- Player on bets on either Red or black

- Wins double the bet money, loss the best is lost

Find:

a) Expected value of earnings X if we place a bet of $3

b) Expected value and standard deviation if we bet $1 each on three rounds

c) compare the two answers in a and b and comment on the riskiness of the two games

Solution:

- Define variable X as the total winnings per round. We will construct a distribution tables for total winnings per round for bets of $3 and $ 1:

- Bet: $3

         X                   -3                          3                              E(X)

        P(X)           1-0.48 = 0.5135      18/37 = 0.4864    3*(.4864-.52135) = -0.08

-The s.d(X) = sqrt(9*(0.5135 + 0.4864) - (-0.08)^2) = 3.0

- Bet: $1

         X                   -1                          1                          E (X)      

        P(X)        1-0.48 = 0.5135      18/37 = 0.4864   1*(.4864-.5135) = -0.0271

-The s.d(X) = sqrt(1*(0.5135 + 0.4864) - (-0.0271)^2) = 0.999

- The expected value for 3 rounds is:

                       E(X_1 + X_2 + X_3) = E(X_1) + E(X_2) + E(X_3)

- The above X winnings are independent from each round, hence:

                       E(3*X_1) = 3*E(X_1) = 3*(-0.0271) = -0.0813

- The standard deviation for 3 rounds is:

              sqrt(Var(X_1 + X_2 + X_3)) = sqrt(Var(X_1) + Var(X_2) + Var(X_3))

- The above X winnings are independent from each round, hence:

                       sqrt(Var(3*X_1)) = 3*Var(X_1) = 3*(0.999) = 2.9988

- For above two games are similar with an expected loss of $0.0813 for playing the game and stakes are very high due to high amount of deviation for +/- $3 of winnings.

In the game of European roulette, betting $1 in three different rounds is less risky than betting $3 in one round as it has a lower expected loss and lower variability.

In the game of roulette, the probability of winning (i.e. the ball landing on either red or black) is 18/37, and the probability of losing (the ball landing on green) is 1/37.

(a) When you bet $3 on a single round, the expected value of your winnings is given by (18/37 * $6) - (19/37 * $3) = -$0.081, and the standard deviation is sqrt([$6-$(-0.081)]^2 * 18/37 + [-$3-$(-0.081)]^2 * 19/37) = $2.899.

(b) When you bet $1 in three different rounds, the expected value of your winnings is 3 * [ (18/37 * $2) - (19/37 * $1) ] = -$0.243, and the standard deviation is sqrt(3 * [(($2-$(-0.081))^2 * 18/37) + (($-1-$(-0.081))^2 * 19/37)]) = $1.578.

(c) If we compare the two games, it is clear that betting $1 in three different rounds decreases both the expected loss and the variability of the results, indicating that the latter game is less risky.

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Answer:

Paul should trim both hedges and Jared should mow both lawns.

Each neighbor would save 30 minutes per week

Step-by-step explanation:

The time each neighbor takes to finish each task is presented below:

[tex]\begin{array}{ccc}&Paul&Jared\\Mow&120&30\\Trim&90&60\end{array}[/tex]

Jared is better at mowing the lawn than trimming hedges, while Paul is better at trimming hedges than mowing the lawn. Therefore, the two neighbors could gain by specializing and trading lawn services if Paul were to trim both hedges and Jared were to mow the lawns.

The time saved by each is:

[tex]P = 210 -90-90=30\ min\\J = 90-30-30=30\ min[/tex]

The volume of dump truck is 237.7 feet cubed.

Step-by-step explanation:

Given dimensions are;

Length of dump truck = 12.05 feet

Width of dump truck = 3.86 feet

Height of dump truck = 5.11 feet

Volume = Length * Width * Height

Volume = 12.05 * 3.86 * 5.11

Volume = 237.681 ft³

Rounding off to nearest tenth

Volume = 237.7 ft³

The volume of dump truck is 237.7 feet cubed.

Keywords: volume, multiplication

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A non responses is a failure to reply something and is a condition that is not responding.

There exists various factors that can create this effect, for example: type of survey, bad questions, un-probabilistic sample, etc.

Hence the options E and F are correct.

Learn more about the some solutions to​ nonresponse.

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To tackle nonresponse in surveys, strategies such as reducing undercoverage, using stratified sampling, changing the wording of questions, offering incentives, reducing interview error, and attempting callbacks can be effective. These methods help enhance response rates and the reliability of survey data.

Tackling nonresponse in surveys is crucial for ensuring accurate and reliable data. Here are some effective solutions:

→ Reduce Undercoverage: By ensuring the survey reaches all relevant subpopulations, you can minimize the chances of missing out on certain groups.

→ Use Stratified Sampling: This method can enhance response rates by making sure each subgroup is adequately represented.

→ Change Wording of Questions: Making questions clearer and more straightforward can increase the likelihood of responses.

→ Offer Rewards and Incentives: Providing incentives can motivate participants to complete the survey.

→ Reduce Interview Error: Training interviewers to minimize bias and errors can improve response quality.

→ Attempt Callbacks: Following up with nonrespondents can help in obtaining more responses.

These methods are essential to improve response rates and, consequently, the accuracy of survey results.

Answer:

The area of the parallelogram is [tex]A=\sqrt{26}[/tex].

Step-by-step explanation:

Let's rewrite these two vectors:

[tex]u=i-2j+2k[/tex]

[tex]v=0i+3j-k[/tex]    

Let's recall that the area of the parallelogram is the magnitude of the cross product between these vectors.            

We can use the Determinant method to find it.        

[tex]u \times v=\left[\begin{array}{ccc}i&j&k\\1&-2&2\\0&3&-1\end{array}\right] = i((-2)*(-1)-2*3)-j(1*(-1)-2*0)+k(1*3-(-2)*0)=i(2-6)-j(-1)+k(3)=-4i+j+3k[/tex]

Now, the magnitude is the square root of each component squared. It will be:

[tex]|u \times v|=\sqrt{(-4)^{2}+(1)^{2}+(3)^{2}}=\sqrt{16+1+9}=\sqrt{26}[/tex]

Therefore the [tex]A=\sqrt{26}[/tex].      

I hope it helps you!

The area of the parallelogram formed by vectors u = i - 2j + 2k and v = 3j - k is 3[tex]\sqrt{10}[/tex] square units, calculated using the cross product formula.

To find the area of the parallelogram with adjacent sides u and v, where:

u = i - 2j + 2k

v = 3j - k

We can use the cross product of u and v to calculate the area. The magnitude of the cross product represents the area of the parallelogram formed by these vectors.

The cross product of two vectors u and v is given by:

u x v = |u| * |v| * sin(θ) * n

Where:

|u| and |v| are the magnitudes of u and v, respectively.

θ is the angle between u and v.

n is the unit vector perpendicular to the plane formed by u and v.

First, let's calculate the magnitudes of u and v:

|u| = [tex]\sqrt{(1^2 + (-2)^2 + 2^2)}[/tex] = [tex]\sqrt{(1 + 4 + 4)}[/tex] = [tex]\sqrt{9}[/tex]= 3

|v| = [tex]\sqrt{(0^2 + 3^2 + (-1)^2)}[/tex] = [tex]\sqrt{(0 + 9 + 1)}[/tex] = [tex]\sqrt{10}[/tex]

Now, let's find the angle θ between u and v. We can use the dot product formula:

u · v = |u| * |v| * cos(θ)

Since u · v = 0 (they are orthogonal), we have:

0 = 3 * [tex]\sqrt{10}[/tex] * cos(θ)

cos(θ) = 0

This means θ is 90 degrees (π/2 radians).

Now, we can calculate the area using the cross product formula:

Area = |u x v| = |u| * |v| * sin(θ)

Area = 3 * [tex]\sqrt{10}[/tex] * 1 (sin(π/2) = 1)

Area = 3[tex]\sqrt{10}[/tex] square units

So, the area of the parallelogram formed by the vectors u and v is 3[tex]\sqrt{10}[/tex]square units.

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Complete question below :

Given two vectors u = 2i - 3j + k and v = i + 4j - 2k, calculate the area of the parallelogram formed by these vectors.

Answer:

28 sandwiches

Step-by-step explanation:

If Jack takes 7 minutes to make a pie, the time that would take Jack to produce 4 pies is:

[tex]t=4*7=28\ minutes[/tex]

Jack would be willing to trade away the amount of sandwiches he is able to produce in 28 minutes to get 4 pies from Rodger. In 28 minutes, the number of sandwiches Jack can produce is:

[tex]S=1*28=28\ sandwiches[/tex]

Jack would be willing to trade away 28 sandwiches for 4 pies.

Answer:

The solution to the differential equation

y' = (7cos5x)/(8 - 3y); y(0) = 3

is

16y - 3y² = 70sin5x + 21

Step-by-step explanation:

y' = (7cos5x)/(8 - 3y)

This can be written as

dy/dx = (7cos5x)/(8 - 3y)

Separate the variables

(8 - 3y)dy = (7cos5x)dx

Integrate both sides

8y - (3/2)y² = 35sin5x + C

Applying the initial condition y(0) = 3

8(3) - (3/2)(3)² = 35sin(5(0)) + C

24 - (27/2) = 0 + C

C = 21/2

Therefore,

8y - (3/2)y² = 35sin5x + 21/2

Or

16y - 3y² = 70sin5x + 21

Answer:

She used the simple random sampling technique because there was no condition attached to the samples she took.

Step-by-step explanation:

we have basically four types of sampling

1.Simple random sampling.

2.Systematic sampling.

3.Stratified sampling.

4.Cluster sampling.

simple Random sampling: is a sampling technique where every item in the population has an even chance and likelihood of being selected in the sample.

Answer:

3.945 years

Step-by-step explanation:

To answer this problem, one must know that Beethoven has composed 32 piano sonatas, 9 symphonies and 5 piano concertos.

The number of different arrangements that can be made by playing a sonata, then a symphony and then a piano concerto is:

[tex]n=32*9*5=1,440[/tex]

If a year has 365 days, the number of years that this daily policy could be continued before exactly the same program would have to be repeated is:

[tex]y=\frac{1,440}{365}=3.945\ years[/tex]

Answer:

B.) square root of 2

E.) -0.54

G.) 5/3

H.) 1.29

Step-by-step explanation:

A probability of an event is how likely the event is to occur. It is always positive values, between 0% and 100%, or as decimals, between 0 and 1.

A.) 1

Can be a probability

B.) square root of 2

The square root of 2 is 1.41. 1.41 is higher than 1, so square root of 2 cannot be a probability

C.) 0

Can be a probability

D.) 0.04

0.04 = 4%

Can be a probability

E.) -0.54

Negative values cannot be probabilities

F.) 3/5

3/5 = 0.6 = 60%

Can be a probability

G.) 5/3

5/3 = 1.67

Higher than 1, so cannot be a probability

H.) 1.29

Higher than 1, cannot be a probability

Answer:

For the exhange of ions and gases such as potassium, calcium and sodium, large fish larva have gills even in early stages. One other main reason is that large fish arvas mostly have circulating red cells after hatching phase while small fish larvae doesn't. So, for these reasons small fish larvae have no gills until later in life.

Answer:

it is not affected by a change of units

Step-by-step explanation:

Since the correlation coefficient has no dimensions, it is not affected by a change of units. Then it will remain the same after the conversion

In fact, the linear correlation coefficient ρ ,where

ρ = Cov (X,Y) / (σx*σy)

then the units [ ] of ρ are

[ρ] = [ Cov (X,Y) ] / [σx]*[σy] = σ²/σ² = 1 → dimensionless

is more useful than using covariance [ Cov (X,Y) ]  , since dividing by the standard deviations eliminates the units and standardise the variable

You can convert the given normal distribution to standard normal distribution and then use the z tables to find the needed probabilities.

Rounding to one places of decimal, we get the answers as:

If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.

If we have

[tex]X \sim N(\mu, \sigma)[/tex]

(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])

then it can be converted to standard normal distribution as

[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]

Using the z scores will help to find the probabilities from the z tables(available online).

Let for the given test, the test scores be tracked by a random variable X, then by the given data, we have:

[tex]X \sim N(530, 119)[/tex]

The needed probabilities are

Converting the distribution to standard normal variate, the probabilities become

[tex]Z = \dfrac{X - 530}{119}\\\\Z \sim N(0, 1)[/tex]
The probabilities convert to

a) [tex]P( 411 < X < 649 ) = P(X < 649) - p(X < 411)\\[/tex]

[tex]P(\dfrac{411 - 530}{119} < Z < \dfrac{ 649 - 530}{119}) = P(-1 < Z < 1) = P(Z < 1) - P(Z < -1)[/tex]

(Know the fact that in continuous distribution, probability of a single point is 0, so we can write [tex]P(Z < a) = P(Z \leq a)[/tex] )

Also, know that if we look for Z = a in z tables, the p value we get is [tex]P(Z \leq a) = p \: value[/tex]

The p value at Z = 1 is 0.8413 and at Z = -1 is 0.1587,

Thus, [tex]P(411 < X < 649) = P(Z < 1) - P(Z < -1) = 0.8413 - 0.1587 = 0.6826[/tex]

b) [tex]P(X < 411) + P(X > 649) = 1 - P(411 \leq X \leq 649) = 1 - P(411 < X < 649)\\\\P(X < 411) + P(X > 649) = 1 - 0.6826 = 0.3174[/tex]

c) [tex]P(X > 768) = 1 - P(X \leq 768)[/tex]

[tex]P(X > 768) = 1 - P(Z < \dfrac{768 - 530}{119}) = 1 - P(Z < 2) = 1 - 0.9772 = 0.0228[/tex]

Rounding to one places of decimal, we get the answers as:

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Answer:

The balance will be $7,577.03.

Step-by-step explanation:

The compound interest formula is given by:

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

Where A is the amount of money, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per unit t and t is the time the money is invested or borrowed for.

In this problem, we have that:

[tex]P = 7000, r = 0.04[/tex]

Semianually means twice a year, so [tex]n = 2[/tex]

We want to find A when [tex]t = 2[/tex].

So

[tex]A = P(1 + \frac{r}{n})^{nt}[/tex]

[tex]A = 7000(1 + \frac{0.04}{2})^{2*2}[/tex]

[tex]A = 7577.03[/tex]

The balance will be $7,577.03.

Answer: [tex]P(A) = 0.23[/tex]

Step-by-step explanation:

Given :

[tex]P(A/B) = 0.2[/tex]

[tex]P(A/B^{1})=0.3[/tex]

[tex]P(B)= 0.7[/tex]

[tex]P(A) = ?[/tex]

From rules of  probability :

[tex]P(A) = P(AnB) + P(A n B^{1})[/tex] ........................... equation *

[tex]P(A n B)[/tex] can be written as [tex]P(A/B)[/tex] x [tex]P(B)[/tex]

Also , [tex]P(A/B^{1})[/tex] can be written as  [tex]P(A/B^{1})[/tex] x [tex]P(B^{1})[/tex]

substituting into equation * , we have

[tex]P(A) = P(A/B)[/tex][tex]P(B) + P(A/B^{1})P(B^{1})[/tex]

since P(B) = 0.7, then [tex]P(B^{1}) = 1 - P(B) = 0.3[/tex]

so , substituting each values , we have

[tex]P(A) = (0.2)(0.7) + (0.3)(0.3)[/tex]

[tex]P(A) = 0.14 + 0.09[/tex]

[tex]P(A) = 0.23[/tex]

Answer:

a.) A= [RRR, LLL, SSS]

b.) B=[RLS, RSL, SRL, SLR, LRS, LSR]

c.) C=[RRL, RRS, RSR, RLR, SRR, LRR]

e.) D'= C' = [RRR, LLL, SSS, RLS, RSL, SRL, SLR, LRS, LSR, LLR, LLS, LSL, LRL, SLL, RLL, SSL, SSR, SLS, SRS, RSS, LSS]

Step-by-step explanation:

If three consecutive vehicles are considered with their direction as R, L and S, then

a.) when the three vehicles go in same direction, A= [RRR, LLL, SSS]

b.) When the three vehicles take the different direction, B=[RLS, RSL, SRL, SLR, LRS, LSR]

c.) when exactly two vehicles turn right, C=[RRL, RRS, RSR, RLR, SRR, LRR]

d.) repeated question of C above.

e.) D'= C' = [RRR, LLL, SSS, RLS, RSL, SRL, SLR, LRS, LSR, LLR, LLS, LSL, LRL, SLL, RLL, SSL, SSR, SLS, SRS, RSS, LSS]

Answer:

0.75 or 75%

Step-by-step explanation:

Let R be the dominant allele for rolling the tongue and r be the recessive allele for not rolling the tongue. If both Dave and Nancy can roll their tongues and had a parent that could not, they both have a heterozygous genotype (Rr).

The sample space for the genotype of their first born child is:

S={RR, Rr, rR, rr}

Only the homozygous recessive genotype rr makes it so that their child is not able to roll his tongue. Therefore, the probability that their first born child will be able to roll his tongue is:

[tex]P=\frac{3}{4}=0.75=75\%[/tex]