Answer with explanation:
It is given that , n is Odd integer.
If , n is odd, then it can be Written as with the help of Euclid division lemma
→ n= 2 p +1, as 2 p is even , and adding 1 to it converts it into Odd.
As Euclid lemma states that for any three integers, a ,b and c ,when a is divided by b, gives quotient c and remainder r , then it can be written as:
a= b c+r, →→0≤r<b
Now, n² will be of the form
(2 p +1)²=(2 p)²+2 × 2 p×1+ (1)²
=4 p²+4 p +1
⇒Multiplying any positive or negative Integer by 4, gives Even integer and sum or Difference of two even integer is always even.
So, 4 p²+4 p, will be an even term.But Adding , 1 to it converts it into Odd Integer.
Hence, if n is an Odd number then , n² will be also odd.
An ISU Computer Science shirt is sold in 6 colors, 5 sizes, striped or solid, and
long sleeve or short sleeve. (a) How many different shirts are being sold? (b) What if
the black and yellow shirts only come in short-sleeve and solid?
a)
[tex]6\cdot5\cdot2\cdot2=120[/tex]
b)
[tex]4\cdot5\cdot2\cdot2 +2\cdot5\cdot1\cdot1=80+10=90[/tex]
The mean per capita income is 24,787 dollars per annum with a variance of 169,744.What is the probability that the sample mean would differ from the true mean by greater than 42 dollars if a sample of 412 persons is randomly selected? Round your answer to four decimal places.
The question asks for the probability that the sample mean of income differs from the true mean by over 42 dollars for a sample size of 412. We use the Central Limit Theorem to approach this problem, calculating the standard error and Z-score to refer to the standard normal distribution to find the associated probability.
Explanation:This question pertains to the field of statistics, more specifically, the Central Limit Theorem. The Central Limit Theorem says that if we take many samples from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population distribution. We can use this theorem to calculate a probability related to the sample mean.
In this case, the population mean (μ) is a per capita income of 24,787 dollars and the population variance (σ²) is 169,744 dollars. We're asked to find the probability that the sample mean would differ from the true mean by more than 42 dollars if a sample of 412 persons is randomly selected.
The standard error of the sample mean is calculated by σ / sqrt(n), where σ is the standard deviation (sqrt(σ²)), and n is the sample size (412). After finding the standard error, we will calculate the Z-score of 42, which is the number of standard errors 42 is away from the mean. Calculating the Z-score is achieved by z = (X - μ) / SE, where X is the value of 42.
We can refer the calculated Z-score to the standard normal distribution to find the associated probability. However, since we are looking for the probability of a difference greater than 42, we want the probabilities in the tails of the distribution beyond our calculated Z-score, so it would be 1 minus this value.
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A multiple choice test contains 10 questions, each with 3 possible answers (of which only one is correct). If a student answers each question by rolling a die and choosing the first answer if the die shows 1 or 2, the second answer if the die shows 3 or 4, or the third answer if the die shows 5 or 6, what is the probability that the student will get exactly 6 correct answers? more than 6 correct answers?
Answer:
1. The probability that the student will get exactly 6 correct answers is [tex]\frac{1120}{19683}[/tex].
2. The probability that the student will get more than 6 correct answers is [tex]\frac{43}{2187}[/tex].
Step-by-step explanation:
From the given information it is clear that
The total number of equations (n) = 10
The probability of selecting the correct answer (p)= [tex]\frac{1}{3}[/tex]
The probability of selecting the incorrect answer (q)= [tex]1-p=1-\frac{1}{3}=\frac{2}{3}[/tex]
According to the binomial distribution, the probability of selecting r items from n items is
[tex]P=^nC_rp^rq^{n-r}[/tex]
where, p is probability of success and q is the probability of failure.
The probability that the student will get exactly 6 correct answers is
[tex]P(r=6)=^{10}C_6(\frac{1}{3})^6(\frac{2}{3})^{10-6}[/tex]
[tex]P(r=6)=210(\frac{1}{3})^6(\frac{2}{3})^{4}=\frac{1120}{19683}[/tex]
Therefore the probability that the student will get exactly 6 correct answers is [tex]\frac{1120}{19683}[/tex].
The probability that the student will get more than 6 correct answers is
[tex]P(r>6)=^{10}C_7(\frac{1}{3})^7(\frac{2}{3})^{10-7}+^{10}C_8(\frac{1}{3})^8(\frac{2}{3})^{10-8}+^{10}C_9(\frac{1}{3})^9(\frac{2}{3})^{10-9}+^{10}C_{10}(\frac{1}{3})^{10}(\frac{2}{3})^{10-10}[/tex]
[tex]P(r>6)=^{10}C_7(\frac{1}{3})^7(\frac{2}{3})^{3}+^{10}C_8(\frac{1}{3})^8(\frac{2}{3})^{2}+^{10}C_9(\frac{1}{3})^9(\frac{2}{3})^{1}+^{10}C_{10}(\frac{1}{3})^{10}(\frac{2}{3})^{0}[/tex]
[tex]P(r>6)=120\times \frac{8}{59049}+45\times \frac{4}{59049}+10\times \frac{2}{59049}+1\times \frac{1}{59049}=\frac{43}{2187}[/tex]
Therefore the probability that the student will get more than 6 correct answers is [tex]\frac{43}{2187}[/tex].
Two friends visited Washington D.C. for the weekend. Person 1 rode the subway one stop, 5 times at the peak fare price and four times at the off peak fare price for a total cost of $17.40. Person 2 rode with Person 1, 2 times at the peak fare price and three times at the off peak fair price for a total cost of $9.20. How much was the peak fare price?
Answer:
The peak fare price is $2,2
Step-by-step explanation:
Considering
peak fare price= xoff peak fare price = yPerson 1
5x + 4y= 17,40
Person 2
2x + 3y= 9,20
Isolating the x
2x= 9,20-3y
x=(9,20-3y)/2=4,6 -1,5y
x=4,6 -1,5y
Replacing in the other equation
5 (4,6 -1,5y) + 4y=17,40
23-7,5y+4y=17,40
23-17,40=7,5y-4y
5,6=3,5y
y=1,60 (this is the off peak fare)
x= 4,60-1,5y
x= 4,60-1,5*1,60
x=2,2 (this is the peak fare)
Final answer:
To determine the peak fare price, we formed a system of linear equations based on the given data. By using the elimination method, we solved the system to find that the peak fare price is $2.20.
Explanation:
We have a system of linear equations representing the total costs of peak and off-peak fares for two friends using public transportation in Washington D.C. Let's denote the peak fare as P and the off-peak fare as Op. According to the problem, we are given the following two equations based on the trips taken and the total cost paid by each person:
5P + 4Op = $17.40 (Person 1)
2P + 3Op = $9.20 (Person 2)
To find the peak fare price, we need to solve this system of equations. We can use a variety of methods, including substitution or the elimination method.
Step 1 - Use the Elimination Method:
Multiply the second equation by 2 to align the Op terms:
4P + 6Op = $18.40 (after multiplying by 2)
Step 2 - Subtract the resulting equation from the first person's total cost:
5P + 4Op - (4P + 6Op) = $17.40 - $18.40
P - 2Op = -$1.00
Step 3 - Solve for P:
Isolate P by adding 2Op to both sides:
P = 2Op - $1.00
Now, substitute P into the second equation from Person 2's total cost:
2(2Op - $1.00) + 3Op = $9.20
4Op - $2.00 + 3Op = $9.20
7Op - $2.00 = $9.20
Add $2.00 to both sides to get 7Op alone:
7Op = $11.20
Divide by 7 to find the off-peak fare:
Op = $11.20 / 7
Op = $1.60
Use the value of Op to find P:
P = 2($1.60) - $1.0
P = $3.20 - $1.00
P = $2.20
The peak fare price is $2.20.
Question 5: What is the radius of the circle with general form x2+y2-4y=21?
Question 5 options:
√21
4.58
5
25
Answer:
r=5 units
Step-by-step explanation:
we have
[tex]x^{2}+y^{2} -4y=21[/tex]
Convert the equation of the circle in standard form
Complete the square twice. Remember to balance the equation by adding the same constants to each side
[tex]x^{2}+(y^{2} -4y+4)=21+4[/tex]
[tex]x^{2}+(y^{2} -4y+4)=25[/tex]
Rewrite as perfect squares
[tex]x^{2}+(y-2)^{2}=25[/tex]
center (0,2)
radius r=5 units
If a company employed 50 people in 1995, and tripled their employment by 2005, how many total people would be employed if there are 40% more employees than in 2005 by 2015?
Answer:
210
Step-by-step explanation:
In 2005 the employee was tripple of 1995 which is 50×3=150. So number of employees in 2005=150.
In 2015 number of employees were 40% of employees in 2005.
40% of 150 =150×40÷100 = 60,
Therefore number of employees in 2015 =150+60= 210 employees.
Answer:
210 employees in 2015
Step-by-step explanation:
In order to solve this we first have to calculate the number of employees that there were in 2005, since the problem says that they tripled their employees from 1995, that meas three times 50, meaning 150 employees in 2005, if by 2015 the number of employees had increase in 40% we just do a simple rule of three, 150 being the 100% and trying to calculate the 140%:
[tex]\frac{150}{100}= \frac{x}{15} \\x=\frac{150*15}{100}\\ x= 210[/tex]
So now we know that there were 210 employees in 2015.
Huck and Jim are waiting for a raft. The number of rafts floating by over intervals of time is a Poisson process with a rate of λ = 0.4 rafts per day. They agree in advance to let the first raft go and take the second one that comes along. What is the probability that they will have to wait more than a week? Hint: If they have to wait more than a week, what does that say about the number of rafts in a period of 7 days?
Answer: 0.0081
Step-by-step explanation:
Let X be the number of rafts.
Given : The mean number of rafts floating : [tex]\lambda=0.4[/tex] rafts per day .
Then , for 7 days the number of rafts = [tex]\lambda_1=\lambda\times 7=0.4\times7=2.8[/tex] rafts per day .
The formula to calculate the Poisson distribution is given by :_
[tex]P(X=x)=\dfrac{e^{-\lambda_1}\lambda_1^x}{x!}[/tex]
Now, the probability that they will have to wait more than a week is given by :-
[tex]P(X>7)=1-P(X\leq7)\\\\=1-(P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(5)+P(6)+P(7))\\\\=1-(\dfrac{e^{-2.8}2.8^{0}}{0!}+\dfrac{e^{-2.8}2.8^{1}}{1!}+\dfrac{e^{-2.8}2.8^{2}}{2!}+\dfrac{e^{-2.8}2.8^{3}}{3!}+\dfrac{e^{-2.8}2.8^{4}}{4!}+\dfrac{e^{-2.8}2.8^{5}}{5!}+\dfrac{e^{-2.8}2.8^{6}}{6!}+\dfrac{e^{-2.8}2.8^{7}}{7!})\\\\=1-0.991869258012=0.008130741988\approx0.0081[/tex]
Hence, the required probability : 0.0081
Find the intersection and union of the sets A and B given below:
A = { a, m, w, u, 7, y } , B = { w, y, g, 7}
Answer:
Intersection: {w, 7, y}
Union: {a, m, w, u, 7, y, g}
Step-by-step explanation:
The intersection of two or more sets are the elements they have in common, that means the elements that are in all the sets at the same time.
For example, "a" is in A but not in B, that's why is not in the intersection. On the other hand, "w" is in both sets, so it's in the intersection.
The union are all the elements that are in one set or the other, but we don't add the element twice if it's in both sets.
For example, "a" is in A so we add it to the union. "w" is in A so we add it to the union but it's also in B, we don't add it again because it is already in the union.
Susan wants to make 2 square flags to
sell at a crafts fair. The fabric she wants
to buy is 6 meters wide. She doesn't
want any fabric left over. What's the
least amount of fabric she should buy?
A complex number, represented by z = x + iy, may also be visualized as a 2 by 2 matrix
(x y
-y x)
(a) Verify that addition and multiplication of complex numbers de ned via matrix opera-
tions are consistent with the usual addition and multiplication rules.
(b) What is the matrix representation corresponding to (x + iy)^-1?
Answer:
Step-by-step explanation:
A) Suppose that we have the complex numbers
[tex]z= x + iy \quad \text{and} \quad \\\\ \tilde{z}=\tilde{x} + i \tilde{y}[/tex]
Remember that to sum complex numbers, we sum the real parts of the two numbers to get the real part and the imaginary parts of the two numbers to get the imaginary part. Hence,
[tex]z+\tilde{z} = (x + i y) + (\tilde{x} + i \tilde{y}) = (x + \tilde{x})+i (y+\tilde{y})[/tex]
On the other hand, if we sum the matrix visualizations of [tex]z \quad \text{and} \quad \tilde{z}[/tex] we get
[tex]\left[\begin{array}{cc}x &y\\-y&x\end{array}\right] + \left[\begin{array}{cc}\tilde{x}&\tilde{y}\\ -\tilde{y}&\tilde{x}\end{array}\right] = \left[\begin{array}{cc}x + \tilde{x}& y + \tilde{y}\\-(y+\tilde{y})&x+\tilde{x}\end{array}\right][/tex]
which is the matrix visualization of [tex]z + \tilde{z}[/tex].
To multiply two complex numbers, we use the distributive law to multiplly and then separete the real part from the imaginary part
[tex]z \cdot \tilde{z}= (x + iy) \cdot (\tilde{x} + i \tilde{y})=(x \tilde{x} + i x \tilde{y} + i \tilde{x} y - y\tilde{y} ) = (x\tilde{x}-y\yilde{y})+i(x\tilde{y}+\tilde{x}y)[/tex]
Again, if we multiply the matrix visualizations of [tex]z \quad \text{and} \quad \tilde{z}[/tex] we get
[tex]\left[\begin{array}{cc}x&y\\-y&x\end{array}\right]\left[\begin{array}{cc}\tilde{x}&\tilde{y}\\-\tilde{y}&\tilde{x}\end{array}\right] = \left[\begin{array}{cc}x\tilde{x}-y\tilde{y}&x\tilde{y}+y\tilde{x}\\-y\tilde{x}-x\tilde{y}&x\tilde{x}-y\tilde{y}\end{array}\right][/tex]
which is the matrix viasualization of [tex]z\cdot\tilde{z}.[/tex]
B) Since the usual matrix operations are consisten with the usual addition and multiplication rules in the complex numbers, we can use them to find the multiplicative inverses of a complex number [tex]z=x+iy[/tex].
We are looking for the complex number [tex]z^{-1}=(x+iy)^{-1}[/tex] which in terms of matrices is equivalent to find the matrix
[tex]\left[\begin{array}{cc}x&y\\-y & x\end{array}\right]^{-1}= \dfrac{1}{x^{2}+y^{2}} \left[\begin{array}{ccc}x&-y\\y&x\end{array}\right][/tex]
Hence,
[tex]z^{-1}=\dfrac{1}{x^2 +y^2} (x-iy)=\dfrac{1}{|z|^2}(x-iy)[/tex]
Ann wants to mix a tomato sauce that is 17% sugar with a sauce that is 30% sugar to obtain 2.6 quarts of a tomato sauce that is 24% sugar. How much of each should she mix?
Answer:
she should mix 1.2 quarts of sauce with 17% sugar with 1.4 quarts of sauce with 30% sugarStep-by-step explanation:
X= quarts of sauce # 1 ( 17% of sugar)
Y= quarts of sauce # 2( 30 % of sugar )
General Balance
[tex]X+Y=2.6[/tex] EQ (1)
[tex]Y=2.6-X\\[/tex]
SUGAR BALANCE
[tex]0.17X+0.3Y=2.6*0.24\\[/tex] EQ (2)
replace Y from eq(1) into eq(2)
[tex]0.17X+0.3(2.6-X)=0.624\\0.17X+0.78-0.3X=0.624\\-0.13X=0.624-0.780\\X=\frac{-0.156}{-0.13} \\X= 1.2\\Y=2.6-1.2=1.4[/tex]
Use the formula for the present value of an ordinary annuity or the amortization formula to solve the following problem.
PV=$15,000; i=0.03; PMT=$650; n=?
n= (Round up to the nearest integer.)
Use the formula for the present value of an ordinary annuity or the amortization formula to solve the following problem.
PV=$9,000; PMT=$500; n=35; i=?
i= (Type an integer or decimal rounded to three decimal places as needed.)
Answer:
1. n = 40
2.
Step-by-step explanation:
The ordinary annuity formula can be written as ...
PV = PMT(1 -(1+r)^-n)/r
where PMT is the payment per period, r is the interest rate per period, and n is the number of periods.
This formula can be solved explicitly for n, but not for r. Iterative or other methods can be used to find r.
__
1. Filling in the given information, we have ...
15000 = 650(1 -1.03^-n)/0.03
450/650 = 1 - 1.03^-n . . . . . divide by the coefficient of the stuff in parens
1.03^-n = 4/13 . . . . . . . . . . . solve for the exponential term
-n·log(1.03) = log(4/13) . . . . take logarithms
n = log(13/4)/log(1.03) ≈ 39.87 . . . . . solve for n
n ≈ 40
__
2. We can rewrite the annuity formula to make it be a function of i that is zero at the desired value of i.
f(i) = PV -PMT(1 -(1+i)^-n)/i
If we want i as a percentage, then we can replace i with i/100 and fill in the given values to get ...
f(i) = 9000 -500(1 -(1 +i/100)^-35)/(i/100)
f(i) = 1000(9 -50(1 -(1 +i/100)^-35)/i) . . . . multiply the fraction by 100/100
Since we're seeking a value of f(r) that is zero, we can eliminate the factor of 1000.
f(i) = 9 -50(1 - (1+i/100)^-35)/i
The attached graph shows the solution to f(i)=0 is near i=4.27%. As a decimal rounded to 3 decimal places, this is ...
i ≈ 0.043
In a certain state, 50% of adults indicated that sausage is their favorite pizza. Suppose a simple random sample of adults in the state of size 23 is obtained and the number of adults who indicated that sausage is their favorite pizza was 17. What are values of the parameters n, p, and x in the binomial probability experiment?
Answer:
n = 23
x = 17
p = 0.50
Step-by-step explanation:
For a binomial experiment we have the following variables:
1) Number of trials or Sample size:
The number of trials is represented by n. In the given scenario 23 adults were asked about their favorite pizza, so the number of trials in this will be 23. Thus
n = 23
2) Number of success
The number of success is denoted by x. Number of success indicates that how many trials resulted in the favorable outcome. In the given case, choosing a sausage pizza is a success. Since 17 adults chose the sausage, so
x = 17
3) Probability of success on single trial
This is represented by p. It is stated that 50% adults say sausage is their favorite pizza. So,
p = 50% = 0.50
In a binomial probability experiment, n represents the size of the random sample, p represents the probability of success, and x represents the number of successes. In this given scenario, n=23, p=0.50 and x=17.
Explanation:In a binomial probability experiment, the parameters n, p, and x are designated as follows: 'n' is the size of the random sample which in this case is 23. 'p' is the probability of success on a single trial, here it is the percentage of adults who indicated that sausage is their favorite pizza, which in decimal form is 0.50. 'x' is the number of successes, in this scenario, the number of adults in the sample of 23 who prefer sausage on their pizza, which is 17. So, in this experiment, n=23, p=0.50 and x=17. Success in this context is defined as an individual person preferring sausage on their pizza.
To ensure the binomial experiment is valid, and can be approximated by a normal distribution, the quantities np and nq (where q is 1-p, the probability of failure) must both be greater than five (np > 5 and nq > 5). In this case, np = 23*0.50 = 11.5 and nq = 23*0.50 = 11.5, thus the experiment is valid.
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Half of the students in an International School are Americans. One third of the remaining students are Europeans, and the rest are Australians. If there are 150 Australians, how many students are in the school?
Answer:
450 students
Step-by-step explanation:
Let s = number of students
1/2 are Americans 1/2s are Americans
s -1/2s = number of students left
1/2 s = number of students left
1/3 of the students left are Europeans
1/3 (1/2s) = 1/6s are Europeans
The rest are Australians = 150
Students = Americans + Europeans + Australians
s = 1/2s + 1/6s + 150
Combine like terms
s = 1/2*3/3 *s + 1/6s +150
s = 3/6s + 1/6s +150
s = 4/6s + 150
Subtract 4/6s from each side
s -4/6s = 150
6/6s - 4/6s = 150
2/6s = 150
1/3s = 150
Multiply each side by 3
3* 1/3s = 150*3
s = 450
There are 450 students
Answer:
There are 450 students.
Step-by-step explanation:
Let's call
s = total number of students
am = American students
eu = European students
au = Australian students
Half of the students in an International School are Americans. Then,
1/2 s = am [1]
One third of the remaining students are Europeans.
The remaining students are s - 1/2 s = 1/2 s
1/3 × (1/2 s) = 1/6 s = eu [2]
There are 150 Australians.
au = 150 [3]
The total number of students is the sum of Americans, Europeans and Australians.
am + eu + au = s
Replacing [1], [2] and [3].
1/2 s + 1/6 s + 150 = s
2/3 s + 150 = s
1/3 s = 150
s = 450
3. Determine a lower bound for the radius of convergence of series solutions about each given point ro for the given different ial equation (a) y"4y6ry = 0; (b) (z2- 2r -3)"+ry'4y 0; ro 0, o4 ro4, ro= -4, ro = 0.
Answer:
Step-by-step explanation:
37
Evaluate the triple integral_E x^6e^y dV where E is bounded by the parabolic cylinder z = 25 - y^2 and the planes z = 0, x = 5, and x = -5.
To evaluate the triple integral, set up the limits of integration based on the given bounds. The parabolic cylinder and the planes define the limits for x, y, and z. Evaluate the triple integral using appropriate integration techniques.
Explanation:To evaluate this triple integral, we need to set up the limits of integration based on the given bounds of the parabolic cylinder and the planes. The parabolic cylinder is defined by the equation z = 25 - y^2, so we need to find the limits for x, y, and z. Since the plane z = 0 is a boundary, the lower limit of integration for z is 0. The other boundary planes, x = 5 and x = -5, set the limits for x. For y, we need to find the limits based on the parabolic cylinder equation. By rearranging the equation, we have y = ±sqrt(25-z), which gives us the upper and lower limits for y.
Now, we can set up the triple integral as follows:
∫-55 ∫-sqrt(25-z)sqrt(25-z) ∫025-y^2 x^6e^y dzdydx
We can then evaluate this triple integral using the appropriate integration techniques, such as using the power rule and integration by parts for the x^6e^y term.
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To evaluate the triple integral, we need to find the limits of integration for each variable. Given the given bounds, the limits for x and y can be determined, and then the integral can be solved using standard techniques.
Explanation:To evaluate the triple integral, we need to find the limits of integration for each variable.
Given that E is bounded by the parabolic cylinder z = 25 - y^2 and the planes z = 0, x = 5, and x = -5, the limits for y will be -√(25-x^2) to √(25-x^2), the limits for x will be -5 to 5, and the limits for z will be 0 to 25 - y^2.
The integral becomes: ∫∫∫ x^6e^y dz dx dy, with the limits as mentioned above. You can now solve this triple integral using standard integration techniques.
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g A standard deck of playing cards is shuffled and three people each choose a card. Find the probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement, and if the cards are chosen without replacement. (a) The cards are chosen with replacement. (b) The cards are chosen without replacement.
Answer:
The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement is 0.03125 .
The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement is 0.03058
Step-by-step explanation:
Total no. of cards = 52
Spade cards = 13 (Black)
Club cards = 13 (Black)
Heart cards = 13 (Red)
Diamond cards = 13(Red)
Total red cards = 26
With replacement case:
Probability of getting spade on first draw = [tex]\frac{13}{52}[/tex]
Now card is replaced
Probability of getting spade on second draw = [tex]\frac{13}{52}[/tex]
Now card is replaced
Probability of getting red card on third draw = [tex]\frac{26}{52}[/tex]
So, The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement = [tex]\frac{13}{52} \times \frac{13}{52} \times \frac{26}{52}[/tex]
=[tex]\frac{1}{32}[/tex]
= [tex]0.03125[/tex]
Without replacement case:
Probability of getting spade on first draw = [tex]\frac{13}{52}[/tex]
Remaining cards = 51
Remaining spade cards = 12
Probability of getting spade on second draw = [tex]\frac{12}{51}[/tex]
Remaining cards = 50
Probability of getting red card on third draw = [tex]\frac{26}{50}[/tex]
So, The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement = [tex]\frac{13}{52} \times \frac{12}{51} \times \frac{26}{50}[/tex]
= [tex]\frac{13}{425}[/tex]
= [tex]0.03058[/tex]
Hence The probability that the first two cards chosen are spades and the third card is red if the cards are chosen with replacement is 0.03125 . The probability that the first two cards chosen are spades and the third card is red if the cards are chosen without replacement is 0.03058
When the cards are chosen with replacement, the probability is 169/2704. When the cards are chosen without replacement, the probability is 39/884.
Explanation:When the cards are chosen with replacement, the probability that the first two cards chosen are spades and the third card is red can be found by multiplying the probabilities of each event happening. There are 13 spades in a deck of 52 cards, so the probability of choosing a spade on the first draw is 13/52. Since the cards are chosen with replacement, the probability of choosing a spade on the second draw is also 13/52. The probability of choosing a red card on the third draw is 26/52, since there are 26 red cards in the deck. Therefore, the overall probability is (13/52) * (13/52) * (26/52) = 169/2704.
When the cards are chosen without replacement, the probability changes. The probability of choosing a spade on the first draw is 13/52. Since the first card is not replaced, there are now 51 cards left and 12 spades remaining. So the probability of choosing a spade on the second draw is 12/51. Finally, there are 25 red cards left out of 50 total cards, so the probability of choosing a red card on the third draw is 25/50. Therefore, the overall probability is (13/52) * (12/51) * (25/50) = 39/884.
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A golf ball is selected at random from a golf bag. If the golf bag contains 99 type A balls, 88 type B balls, and 44 type C balls, find the probability that the golf ball is not a type A ball.
Answer:
4/7
Step-by-step explanation:
To find the probability, we must first find the total number of golf balls. So 99+88+44 = 231 is our total. Now we must find the number of golf balls that are not a type A ball. This is just the total number of type B balls and type C balls. So 88+44 = 132. So the probability is 132/231. This simplifies to 12/21 or 4/7.
A student took a chemistry exam where the exam scores were mound-shaped with a mean score of 90 and a standard deviation of 64. She also took a statistics exam where the scores were mound-shaped, the mean score was 70 and the standard deviation was 16. If the student's grades were 102 on the chemistry exam and 77 on the statistics exam, then: a. the student did relatively better on the chemistry exam than on the statistics exam, compared to the other students in each class. b. it is impossible to say which of the student's exam scores indicates the better performance. c. the student did relatively better on the statistics exam than on the chemistry exam, compared to the other students in the two classes. d. the student's scores on both exams are comparable, when accounting for the scores of the other students in the two classes. e. the student did relatively the same on both exams
Answer:
wow that school has alot of test don't it lol
Step-by-step explanation:
anyway what's the question
After calculating the Z-scores for the student's performance in each class, we found that she performed closer to the top of her class in statistics. Thus, she did relatively better on the statistics exam compared to the chemistry exam.
Explanation:To determine relative performance in each class, we need to calculate the number of standard deviations the student's score is from the mean (also known as a
Z-score
).
For the chemistry exam, where the mean was 90 and the standard deviation was 64, the student's Z-score is: (102 - 90) / 64 = 0.1875.
For the statistics exam, with a mean of 70 and standard deviation of 16, the student's Z-score is: (77 - 70) / 16 = 0.4375.
Comparing these Z-scores, we can conclude that the student did relatively better in the statistics class because her Z-score (0.4375) was higher there than in her chemistry class (0.1875). This indicates she performed closer to the top of her class in statistics. Therefore, the answer would be selection C: the student did relatively better on the statistics exam than on the chemistry exam, compared to the other students in the two classes.
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Given that f(x)= x(squared)-3x Evaluate f(2+h)-f(2)/h
Answer: The required value of the given expression is (h +1).
Step-by-step explanation: We are given a function f(x) as follows :
[tex]f(x)=x^2-3x~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
We are to evaluate the following :
[tex]E=\dfrac{f(2+h)-f(2)}{h}.[/tex]
Substituting x = 2 + h in equation (i), we get
[tex]f(2+h)\\\\=(2+h)^2-3(2+h)\\\\=(4+4h+h^2)-6-3h\\\\=4+4h+h^2-6-3h\\\\=h^2+h-2[/tex]
and substituting x = 2 in equation (i), we get
[tex]f(2)=2^2-3\times2=4-6=-2.[/tex]
Therefore, the value of expression E can be evaluated as follows :
[tex]E\\\\\\=\dfrac{f(2+h)-f(2)}{h}\\\\\\=\dfrac{(h^2+h-2)-(-2)}{h}\\\\\\=\dfrac{h^2+h-2+2}{h}\\\\\\=\dfrac{h^2+h}{h}\\\\\\=\dfrac{h(h+1)}{h}\\\\=h+1.[/tex]
Thus, the required value of the given expression is (h +1).
Determine whether the geometric series is convergent or divergent. 6 + 5 + 25/6 + 125/36 + ...
If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
The [tex]n[/tex]-th term in the series is 6 multiplied by the [tex](n-1)[/tex]-th power of 5/6:
[tex]a_1=6=6\left(\dfrac56\right)^{1-1}[/tex]
[tex]a_2=5=6\left(\dfrac56\right)^{2-1}[/tex]
[tex]a_3=\dfrac{25}6=6\left(\dfrac56\right)^{3-1}[/tex]
and so on.
[tex]\displaystyle\sum_{n=1}^\infty6\left(\frac56\right)^{n-1}[/tex]
Consider the [tex]N[/tex]-th partial sum,
[tex]S_N=\displaystyle\sum_{n=1}^N6\left(\frac56\right)^{n-1}[/tex]
[tex]S_N=6\left(1+\dfrac56+\cdots+\dfrac{5^{N-2}}{6^{N-2}}+\dfrac{5^{N-1}}{6^{N-1}}\right)[/tex]
Multiplying both sides by 5/6 gives
[tex]\dfrac56S_N=6\left(\dfrac56+\dfrac{5^2}{6^2}+\cdots+\dfrac{5^{N-1}}{6^{N-1}}+\dfrac{5^N}{6^N}\right)[/tex]
and substracting this from [tex]S_N[/tex] gives
[tex]\dfrac16S_N=6\left(1-\dfrac{5^N}{6^N}\right)[/tex]
[tex]S_N=36\left(1-\left(\dfrac56\right)^N}\right)[/tex]
As [tex]N\to\infty[/tex], it's clear that the sum converges to 36.
The geometric series in the question is convergent with a common ratio of 5/6. Using the formula for the sum of an infinite geometric series, the sum of the series is found to be 36.
Explanation:In mathematics, specifically in series, determining whether a geometric series is convergent or divergent is centered around the common ratio value. In terms of this particular series: 6 + 5 + 25/6 + 125/36 + ..., the common ratio is 5/6. Given this common ratio, it's clear that it falls between -1 and 1. Hence, this geometric series is convergent.
Once we establish it is a convergent series, we can calculate its sum using the formula for the sum of an infinite geometric series: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. Inserting the respective values a = 6 and r = 5/6, we get: S = 6 / (1 - 5/6) = 36. Hence, the sum of this infinite geometric series is 36.
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Although not so common now, consider a phone plan that does not have unlimited texting. Your base plan costs you $30 a month and each text sent or received is only 12 cents. Write an equation to describe the total monthly cost of your bill with x text messages and y cost.
In one month 217,000 messages were sent between two brothers in Philadelphia. What was their approximate bill? (This is a true story!)
Answer:
The bill will be $26070.
Step-by-step explanation:
Your base plan costs you $30 a month
And each text sent or received is only 12 cents or $0.12.
Let the messages sent be = x
So, cost for messages become = [tex]0.12x[/tex]
Let total bill cost be = y
Now, total bill will comprise of monthly plan charges plus message charges.
So, equation forms:
[tex]y=30+0.12x[/tex]
Now, given is that In one month 217,000 messages were sent between two brothers in Philadelphia.
So, their approximate bill will be :
[tex]y=30+0.12(217000)[/tex]
=>[tex]y=30+26040[/tex]
=> y = 26070
Therefore, bill will be $26070.
The answer explains how to write an equation for the total monthly cost of a phone plan based on text message usage and calculates the bill for 217,000 messages sent between two brothers in Philadelphia.
Explanation:To write an equation for the total monthly cost of a phone plan with a base cost and a cost per text message:
Let x be the number of text messages sent or received.Let y be the total monthly cost.Using the given information, the equation is: y = $30 + 0.12xFor the specific case of 217,000 messages sent between two brothers:
Substitute x = 217,000 into the equation: y = $30 + 0.12(217,000)Calculate to find their approximate bill.If D is the midpoint of segment AB and AB = 15, what is AD?
Answer: AD = 7.5
Step-by-step explanation: A midpoint is halfway between 2 points, which is AB. AB = 15. To find AD, which is half of the line, divide 15 by 2.
15/2 = 7.5
AD is 7.5
Answer:
AD = 7.5 units
Step-by-step explanation:
It is given in the question, a segment AB having measure = 15 units
If D is the midpoint of the segment AB, then we have to find the measure of segment AD.
Since D is the midpoint of AB then length of segment AD = [tex]\frac{1}{2}\times AB[/tex]
= [tex]\frac{1}{2}\times 15[/tex]
= 7.5 units
Therefore, AD = 7.5 units will be the answer.
Suppose babies born in a large hospital have a mean weight of 4095 grams, and a standard deviation of 569 grams. If 130 babies are sampled at random from the hospital, what is the probability that the mean weight of the sample babies would differ from the population mean by greater than 42 grams? Round your answer to four decimal places.
Answer:
[tex]P =0.3998[/tex]
Step-by-step explanation:
Let [tex]{\displaystyle {\overline{x}}}[/tex] be the average of the sample, and the population mean will be [tex]\mu[/tex]
We know that:
[tex]\mu = 4095[/tex] gr
Let [tex]\sigma[/tex] be the standard deviation and n the sample size, then we know that the standard error of the sample is:
[tex]E=\frac{\sigma}{\sqrt{n}}[/tex]
Where
[tex]\sigma=569[/tex]
[tex]n=130[/tex]
In this case we are looking for:
[tex]P(|{\displaystyle{\overline{x}}}- \mu|>42)[/tex]
This is:
[tex]{\displaystyle{\overline{x}}}- \mu>42[/tex] or [tex]{\displaystyle{\overline{x}}}- \mu<-42[/tex]
[tex]P=P({\displaystyle{\overline{x}}}- \mu>42)+ P({\displaystyle{\overline{x}}}- \mu<-42)[/tex]
Now we get the z score
[tex]Z=\frac{{\displaystyle{\overline{x}}}-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]P=P(z>\frac{42}{\frac{569}{\sqrt{130}}}) + P(z<-\frac{42}{\frac{569}{\sqrt{130}}})[/tex]
[tex]P=P(z>0.8416) + P(z<-0.8416)[/tex]
Looking at the tables for the standard nominal distribution we get
[tex]P =0.1999+0.1999[/tex]
[tex]P =0.3998[/tex]
The functions q and r are defined as follows.
q(x) = -x - 2
r(x) = -2x^2 -2
Find the value of r(q(-4)) .
Answer:
r(q(-4)) = -10
Step-by-step explanation:
q(x) = -x - 2
r(x) = -2x^2 -2
r(q(-4))
First find q(-4)
Let x=-4
q(-4) = -(-4) - 2
q(-4) = +4 -2 = 2
q(-4) =2
We substitute this value in for x in r(x)
r(2) = -2(2)^2 -2
= -2 (4) -2
= -8 -2
= -10
For this case we have the following functions:
[tex]q (x) = - x-2\\r (x) = - 2x ^ 2-2[/tex]
We must find [tex]r (q (x))[/tex]. So:
We substitute [tex]q (x)[/tex]in [tex]r (x).[/tex]
[tex]r (q (x)) = - 2 (-x-2) ^ 2-2[/tex]
Now we substitute[tex]x = -4[/tex]
[tex]r (q (-4)) = - 2 (- (- 4) -2) ^ 2-2\\r (q (-4)) = - 2 (+ 4-2) ^ 2-2\\r (q (-4)) = - 2 (2) ^ 2-2\\r (q (-4)) = - 2 (4) -2\\r (q (-4)) = - 8-2\\r (q (-4)) = - 10[/tex]
Answer:
[tex]r (q (-4)) = - 10[/tex]
Check My Work (No more tries available) Solve the following word problem, rounding dollars to the nearest cent. The Flour Power Bakery makes 280 cherry cheesecakes at a cost of $2.51 each, If a spoilage rate of 30% is anticipated, at what price should the cakes be sold to achieye a 65% markup based on cost? Do not enter units in your answer 10.24 per cheesecake
Answer:
5.92 per cheesecake.
Step-by-step explanation:
The initial number cherry cheesecakes = 280,
Also, the cost of each cheesecakes = $ 2.51,
So, the total cost price = 280 × 2.51 = $ 702.8,
Markup = 65 %,
Thus, the total selling cost = 702.8 + 65% of 702.8 = $ 1159.62,
Now, the spoilage rate = 30 %,
So, the total new number of cheesecakes = 280 - 30% of 280 = 280 - 84 = 196,
Hence, the selling cost of each cheesecake = [tex]\frac{1159.62}{196}[/tex]
[tex]=\$ 5.91642857143[/tex]
[tex]\approx \$ 5.92[/tex]
6. Let's say the countdown for a space shuttle launch has begun. At "T minus 27 hours" (that is, 27 hours before launch), a problem occurs. If the technicians have not fixed the problem by T minus 8 hours, the launch will have to be scratched. How much time do the technicians have to correct the problem?
Answer: the technicians have 19 hours to correct the problem.
Step-by-step explanation:
The countdown started at "T-minus 27 hours" that means that the clock starts ticking backwards and if an hour passes away the lime left for the launch will be 26 hours and the limit the technicians have is T minus 8 hours so the problem can be solved with a simple substraction 27 hours minus 8 hours equals 19 hours.
1.prove that the following three functions are linearly dependent
f1=x^2; f2(x)=1-x^2; f3=2+x^2
Proof:
Given any functions [tex]f_{1}(x),f_{2}(x),f_{3}(x)[/tex] they are linearly dependent if we can find values of [tex]c_{1},c_{2},c_{3}[/tex] such that
[tex]c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0[/tex]
Using the given functions in the above equation we get
[tex]c_{1}f_{1}(x)+c_2f_{2}(x)+c_{3}f_{3}(x)=0\\\\c_{1}x^{2}+c_{2}(1-x^{2})+c_{3}(2+x^{2})=0\\\\\Rightarrow (c_{1}-c_{2}+c_{3})x^{2}+c_{1}+c_{2}+2c_{3}=0[/tex]
This will be satisfied if and only if
[tex]c_1-c_2+c_3=0,c_1+c_2+2c_3=0[/tex]
Solving the equations we get
[tex]c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0[/tex]
Since we have 3 variables and 2 equations thus we will get many solutions
one being if we put [tex]c_3=1[/tex] we get
[tex]c_1+2c_3=-c_2\\\\c_1+c_1+2c_3+c_3=0\\2c_1+3c_3=0\\\\c_1=\frac{-3}{2}\\\\c_2=\frac{-1}{2}[/tex]
Thus we have [tex]c_1=\frac{-3}{2},c_2=\frac{-1}{2},c_3=1[/tex] as one solution. Hence the given functions are linearly dependent.
What does the symbol R indicate in an experimental design? a. observation b. experimental variable C. comparison groups d. random assignment
Answer:
R indicates Random Assignment
Step-by-step explanation:
In designing an experiment, R is the symbol used for assigning the subjects randomly to different groups through randomization. It is called random assignment and It is the option (d) random assignment. It is achieved by using chance procedure ( random number generator, by flipping coin, throwing ). This ensures that all the subjects have equal chance of placing in all the groups.
Option (a) Symbol O is used for Observations. (So option (a) is wrong answer)
Option (b) Symbols x , y are used as Observational variables (So option (b) is wrong answer)
Option (c) Comparison groups is wrong answer as R is not used for that.
The symbol R in experimental design represents random assignment, a technique used to ensure that groups in an experiment are initially equivalent and thereby allow researchers to make causal inferences.
Explanation:In experimental design, the symbol R indicates random assignment. Random assignment is a process used to create initial equivalence between the groups in an experiment, which is crucial for allowing researchers to draw causal conclusions. In this process, participants are assigned to different groups or conditions on a random basis, such as by drawing numbers or using a random number generator.
Random assignment minimizes the effects of lurking variables, ensuring that before the experimental manipulation occurs, the groups can be considered equivalent on various potential confounding factors. When individuals are selected into groups denoted by R, we can generally classify these designs as experimental, as opposed to nonexperimental designs where nonrandom assignment (NR) is indicated.
For example, in a study where the size of tableware is manipulated to see its effect on food consumption, the psychologist would use random assignment to ensure that both treatment groups are comparable prior to the intervention, which is the manipulation of the independent variable.
10. Sarah is planning to fence in her backyard garden. One side of the garden is 34 feet long, another side is 30 feet long, and the third side is 67 feet long.Find the perimeter of Sarah’s garden to determine the amount of fencing material needed.
A.262 ft.
B.68,340 ft.
C.250 ft.
D.131 ft.
Answer:
131ft is the amount of fencing material needed
Step-by-step explanation:
Perimeter is the distance around a shape: we have to sum all the distances
P = d1 + d2 + d3
P = 34 ft + 30 ft + 67 ft = 131 ft
Answer:
D. 131 ft.
Step-by-step explanation:
If Sarah is planning to fence in her backyard garden and one side of the garden is 34 feet long, another side is 30 feet long, and the third side is 67 feet long. The perimeter of Sarah’s garden to determine the amount of fencing material needed is 131 feet.