Working alone John can wash the windows of a building in 2.5 hours Caroline can wash the building windows by her self in 4 hours if they work together how many hours should it take to wash the windows

Therefore, by PMI, we can conclude that 22n+1 +7 is a multiple of 3 for n≥1.

what is multiple  ?

In mathematics, a multiple is a product of an integer and any other integer. In other words, if a and b are integers, and there exists an integer c such that a = b x c, then a is a multiple of b, and we say that b divides a (or that b is a divisor or factor of a).

In the given question,

To prove that 22n+1 + 7 is a multiple of 3 for n≥1 using PMI, we will first establish the base case and then prove the induction step.

Base case:

When n=1, we have:

22n+1 + 7 = 22(1)+1 + 7 = 4+7 = 11

11 is not a multiple of 3. So the statement is not true for n=1.

Induction step:

Assuming that the statement is true for some integer k≥1, we will prove that it is also true for k+1.

So, we need to show that:

22(k+1)+1 + 7 is a multiple of 3

Expanding the expression, we have:

22(k+1)+1 + 7 = 2 x 22k+1 x 2 + 7

= 2 x 22k+2 + 7

We can write 2 as (3-1), and substitute it in the expression:

2 x 22k+2 + 7 = (3-1) x 22k+2 + 7

= 3 x 22k+2 - 22k+2 + 7

Notice that 22k+2 is a multiple of 3 because 22k+1 is even, and 22 is a multiple of 3. Therefore, we have:

3 x 22k+2 - 22k+2 + 7 = 3 x (22k+2) + (-22k+2 + 7)

= 3 x (22k+2) + (-22k+5)

Now, let's consider the expression (-22k+5). We can rewrite it as (-22k+3+2):

(-22k+5) = (-22k+3+2) = -3 x 2k+1 + 2 x 1

Since 2k+1 is odd, we can write it as 2j+1, where j is an integer. Substituting this value in the expression, we have:

-3 x 2k+1 + 2 x 1 = -3 x 2j+2 + 2

= -3 x (2j+1) + 2

= -3 x 2j+1 + (-3+2)

= -3 x 2j+1 - 1

So, we can rewrite the original expression as:

3 x (22k+2) + (-22k+5) = 3 x (22k+2) - 3 x 2j+1 - 1

= 3 x (22k+2 - 2j-1) - 1

Notice that 22k+2 - 2j-1 is an even number, and therefore, it can be written as 2m for some integer m. Substituting this value, we have:

3 x (22k+2 - 2j-1) - 1 = 3 x 2m - 1

= 3m + 3m + 3 - 1

= 3 x (2m+1) + 2

Since 2m+1 is an integer, we can conclude that 22(k+1)+1 + 7 is a multiple of 3.

Therefore, by PMI, we can conclude that 22n+1 +7 is a multiple of 3 for n≥1.

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The answer is True we use observed value not critical value

The answer is True. When computing the effect size, you use the observed value of t in the formula rather than the critical value. The critical value is used to determine statistical significance, while the effect size is calculated using the observed value to measure the strength or magnitude of the relationship between variables.

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The Fido's share in percentage is 470%.

The term "amount" typically refers to the quantity or magnitude of something, usually represented by a numerical value. It is a general term that can be used to describe the size, extent, or measure of a quantity.

"Share" refers to a portion or fraction of a whole or a total.

Based on the above conditions, formulate

4*(19-6)÷(4+6)

on calculating it comes out as 52/11

convert the above fraction in percentage gives the result 470%.

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1. R = {(a, b, c): a, b, c are positive integers and a < b < c} ; 2. S = {(x, y): x is a multiple of y}

For the first relation R, we can see that it is an n-ary relation with n = 3. The condition a < b < c ensures that the triplets (a, b, c) are in increasing order. For example, (1, 2, 3) is a valid element of R, but (3, 2, 1) is not.

For the second relation S, we can see that it is a binary relation with n = 2. The condition x is a multiple of y means that for every pair (x, y) in S, x must be divisible by y. For example, (12, 2) is a valid element of S, but (5, 3) is not.

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By applying L'Hôpital's Rule 100 times or analyzing the degree of the polynomial, you'll find that L equals 0. Since L < 1, by the Ratio Test, this series converges.

To determine whether the series converges, you can use the Ratio Test. For the series Σk=1 to infinity (k100/(k+5)!), apply the Ratio Test by finding the limit as k approaches infinity of the absolute value of the ratio of consecutive terms:

L = lim (k→∞) |( (k+1)100 / (k+6)! ) / ( k100 / (k+5)! )|

Upon simplification, you'll find:

L = lim (k→∞) |(k+1)100 * (k+5)! / (k100 * (k+6)!)|

Cancel out the factorials and further simplify:

L = lim (k→∞) |(k+1)100 / (k100 * (k+6))|

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The slope of the line is - 1.

A line in the cartesian plane, is a particular set of points. Each point in the plane has a pair of cartesian coordinates, one each corresponding to the x and y coordinate respectively. The slope of the line segment joining two points is the change in the y-coordinate, divided by the change in the x-coordinate, as we move from the first point to the second point. The line as a geometric figure is characterized by the property that, the slope of the line segment joining any two distinct points on the line is the same. This slope is then denoted to be the slope of the line.

To calculate the slope of a line, we need two points, on the line. let P,Q be two points on the line. Let [tex](x_p,y_p), (x_q,y_q)[/tex] be the coordinates of the two points. then the slope of the line segment [tex]{PQ}[/tex] is the ratio [tex]\frac{y_p-y_q}{x_p-x_q}[/tex] . This is also equal to the slope of the line containing the points P, Q

In our question the line passes through the points, (1,2) and (-1,4). These points are marked on the line with highlighted dots. So the slope of the line  in the picture is (4-2)/(-1-1) = 2/-2 = -1

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The double line graph for the amount in the River Bank and Town Bank indicates;

13 a. $10,000

13 b. Two year

c. $4,000 per year

A graph of a function shows the relationship that exists between the input and output values of the function.

The graph with the lines that have markings is the graph of the River Bank

The graph with the lines without markings is the graph of the Town Bank

13 a. In the year 2,000, the money in River Bank = $6,000

The money in Town Bank = $4,000

The combined amount in both banks in the year 2,000 is therefore;

Amount  = $6,000 + $4,000 = $10,000

13 b. Town Bank had more money than Rivers Bank in the years; 1998, 2002

Therefore;

Towns Bank had more money that Rivers Bank in 2 years

c. The amount River Bank had between 1998 to 2004 are;

$5,000 + $3,000 + $6,000 + $4,000 + $1,000 + $7,000 + $2,000 = $28,000

The number of years between 1998 and 2004 = 7 years

The mean number of dollars per year River Bank had = $28,000/7 = $4,000 per year

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The integral ∫∫ r √4 −x2−y2da where r = {(x, y) : x2 y2≤ 4, x ≥ 0} conversion to polar coordinates the value of the integral is (4/3)π.

To convert the integral to polar coordinates, we need to express the limits of integration in terms of the polar coordinates.

Recall that in polar coordinates, x = r cosθ and y = r sinθ, where r is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis to the line connecting the origin to the point (x, y).

In this case, the region r is defined by [tex]x^2 + y^2[/tex] ≤ 4 and x ≥ 0. In polar coordinates, this corresponds to the region 0 ≤ r ≤ 2 and 0 ≤ θ ≤ π/2. To see why, note that x ≥ 0 implies 0 ≤ θ ≤ π/2, and [tex]x^2 + y^2 = r^2[/tex], so r ≤ √4 = 2.

So we have:

∫∫ r √4 −x2−y2da = ∫(θ=0 to π/2) ∫(r=0 to 2) r√(4-[tex]r^2[/tex]) dr dθ

To evaluate this integral, we can use the substitution u = 4 - [tex]r^2[/tex], du = -2r dr, which gives:

∫∫ r √4 −x2−y2da = ∫(θ=0 to π/2) ∫(u=4 to 0) -1/2 √u du dθ

Now we can evaluate the inner integral:

∫(u=4 to 0) -1/2 √u du = [-1/3 u^(3/2)](u=4 to 0) = (1/3)(8 - 0) = 8/3

Substituting this back into the original integral, we have:

∫∫ r √4 −x2−y2da = ∫(θ=0 to π/2) (8/3) dθ = (8/3) (π/2 - 0) = (4/3)π

Therefore, the value of the integral is (4/3)π.

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As per th presented informations and the given informations, the cylinder's volume can be found out being approximately 45638.3 cubic meters.

To calculate the volume of a cylinder with a given height and radius, we use the formula V = πr²h, where V is the volume, r is the radius, and h is the height.

putting in the present values, we will be getting:

V = π(9²)(18)

V = π(81)(18)

V = 14526π

Rounding the achieved value to the nearest tenth, we get:

V ≈ 14526π ≈ 45638.3

Therefore, the volume of the cylinder is found out to be  nearly 45638.3 cubic meters.

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The complete question is :

Find the volume of the cylinder. Round your answer to the nearest tenth. Having radius 9m and height 18 m.

Answer:

58

Step-by-step explanation:

Put the numbers in chronological order and find the middle number

0, 0, 38, 47, 58, 65, 78, 99, 227

Helping in the name of Jesus.

Answer:

4 - (2 / 13) = 3.84615385.

But I understand that you don't want to answer.

Step-by-step explanation:

134 pixels total width of an element, where the content is 100 pixels wide, the padding is 10 pixels thick, the border is 2 pixels thick, and the margin is 5 pixels thick.

The total width of the element would be 117 pixels. This is because you need to add the content width of 100 pixels, the left and right padding of 10 pixels each (which makes a total of 20 pixels), the left and right border of 2 pixels each (which makes a total of 4 pixels), and the left and right margin of 5 pixels each (which makes a total of 10 pixels). Adding all these values gives you a total width of 117 pixels.
The total width of an element with a content width of 100 pixels, padding of 10 pixels, border of 2 pixels, and margin of 5 pixels can be calculated as follows:

Total width = Content width + (Padding * 2) + (Border * 2) + (Margin * 2)

Total width = 100 pixels + (10 pixels * 2) + (2 pixels * 2) + (5 pixels * 2)

Total width = 100 pixels + 20 pixels + 4 pixels + 10 pixels

Total width = 134 pixels

So, the total width of the element is 134 pixels.

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When analyzing statistical data, it is important to consider the assumptions underlying the methods used. One such assumption is homogeneity of variance (HOV), which tests whether the variance of two groups is equal.

Why did we not have to test for hov with our paired data?

We do not need to test for the homogeneity of variance (HOV) assumption in paired data because the assumption only applies to independent samples. In paired data, the same individuals are measured twice, and the two sets of measurements are dependent. Therefore, the assumption of equal variances between two groups does not apply, and we do not need to test for it.

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Why did we not have to test for HOV with our paired data?

The derivative dy/dx is (6y - eˣ - yeˣ)/y².

To find dy/dx for the given equation ex/y = 6x - y, you can use implicit differentiation without taking the log.

Given the equation [tex]e^\frac{x}{y}[/tex] = 6x - y, you do not need to take the log of both sides. Instead, start by applying implicit differentiation to both sides with respect to x:

1. Differentiate ex with respect to x: d(eˣ)/dx = ex
2. Differentiate y with respect to x: d(y)/dx = dy/dx
3. Differentiate 6x with respect to x: d(6x)/dx = 6
4. Apply the quotient rule to d([tex]e^\frac{x}{y}[/tex])/dx: d(ex/y)/dx = (y * d(eˣ)/dx - eˣ * d(y)/dx) / y² = (y * eˣ - eˣ * dy/dx) / y²
5. Set the derivatives equal: (y * eˣ - eˣ * dy/dx) / y² = 6 - dy/dx

Now, solve for dy/dx:

6. Multiply both sides by y²: y * eˣ - eˣ * dy/dx = 6y² - y² * dy/dx
7. Rearrange terms: dy/dx * (y² + eˣ) = 6y² - y * eˣ
8. Solve for dy/dx: dy/dx = (6y² - y *eˣ) / (y² + eˣ) = (6y - ex - yeˣ) / y²

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The time of the rth arrival in a Poisson process follows a gamma distribution with parameters r and λ, where λ is the rate parameter.

The density function for the time of the rth arrival is: f(w) = λ^r * w^(r-1) * e^(-λw) / (r-1) where w is the time of the rth arrival. This density function gives the probability density of the time of the rth arrival occurring at a specific time w, given that there have been n arrivals up to time t.

The density function is derived from the fact that the time between successive arrivals in a Poisson process follows an exponential distribution with rate parameter λ, and the time of the rth arrival is the sum of r independent exponential random variables.

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The proof by contradicting of the statement by using appropriate sentences from the list are:

1. Suppose not. That is, suppose there exists positive real numbers r and s such that r + s ≠ √r + √s.
2. Squaring both sides of the equation gives r + s = r + 2√rs + s.
3. Simplifying the equation gives 0 = 2√rs.
4. By the zero product property, at least one of √r or √s equals 0, which implies r or s equals 0.
5. But this is a contradiction because r and s are positive.
6. Thus, we have reached a contradiction and have proved the statement.

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The proof by contradicting of the statement by using appropriate sentences from the list are:

1. Suppose not. That is, suppose there exists positive real numbers r and s such that r + s ≠ √r + √s.
2. Squaring both sides of the equation gives r + s = r + 2√rs + s.
3. Simplifying the equation gives 0 = 2√rs.
4. By the zero product property, at least one of √r or √s equals 0, which implies r or s equals 0.
5. But this is a contradiction because r and s are positive.
6. Thus, we have reached a contradiction and have proved the statement.

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The general solution for the non-homogeneous equation is y(t) = yh(t) + yp(t) = c1 + c2t + c3e^(-0.3726t) + c4e^(0.3726t) + (1/7)t - 1/49.

To determine intervals in which solutions are sure to exist for the differential equation y(4) + 7y''' + 5y = t, we need to look at the characteristic equation, which is r^4 + 7r^3 + 5r = 0. Factoring out an r, we get r(r^3 + 7r^2 + 5) = 0. This gives us two roots: r = 0 and r = -0.3726, approximately. For the homogeneous equation, the general solution is yh(t) = c1 + c2t + c3e^(-0.3726t) + c4e^(0.3726t).

To find particular solutions for the non-homogeneous equation, we need to consider the form of the forcing term t. Since it is a linear function, we can guess a particular solution of the form yp(t) = At + B. Substituting this into the equation, we get 7A - 1 = 0, which means A = 1/7. Then, substituting back in and solving for B, we get B = -1/49. So our particular solution is yp(t) = (1/7)t - 1/49.

Therefore, the general solution for the non-homogeneous equation is y(t) = yh(t) + yp(t) = c1 + c2t + c3e^(-0.3726t) + c4e^(0.3726t) + (1/7)t - 1/49. Since all of the terms in the general solution are continuous and differentiable on the entire real line, solutions are sure to exist for all values of t, and the interval notation for this would be (-∞, ∞).

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The rational number that helps complete the circuit is given as follows:

0.222...

For this problem, we have two options to complete the circuit, as follows:

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The missing statement in the proof is ∠SXT ≅ ∠SVT  by AAS of similar triangles.

1. ∠TXU ≅ ∠TVS  by given

2. ∠STV ≅ ∠UTX by reflex. prop.

3. △STU is an equilateral triangle by  given

4. ST ≅ UT  because sides of an equilat. △ are ≅

5. ∠SXT ≅ ∠SVT by AAS of similar triangle

6. UX ≅ SV by CPCTC

In △SXU and △TVS, we have already proved that ∠TXU ≅ ∠TVS (statement 1) and ∠STU ≅ ∠VTS (statement 2).

Also, we know that ST ≅ UT (statement 4) and so SX ≅ TV (by subtracting UX from both sides).

Therefore, we have two pairs of congruent angles and a pair of congruent sides that are included between these angles which satisfies the AAS congruence criterion, and so we can conclude that △SXU ≅ △TVS.

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A finite set is a set that has a specific number of elements (as opposed to an infinite set, which has an infinite number of elements).

A subset is a set that contains some (or all) of the elements of another set. For example, if A = {1, 2, 3, 4}, then {1, 2} is a subset of A.

The power set of a set is the set of all subsets of that set. So, if A = {1, 2, 3}, then the power set of A is the set { {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }, which contains all possible combinations of the elements of A. Note that the power set always includes the empty set and the set itself.

Now, let's prove the statement |s| = 2|a|.

To do this, we need to show that the number of elements in s (the power set of a) is equal to 2 to the power of the number of elements in a.

First, let's think about how many subsets a finite set with n elements has.

For each element in the set, there are two choices: include it in a particular subset, or don't include it. So, for one element, there are 2 choices. For two elements, there are 2 x 2 = 4 choices. For three elements, there are 2 x 2 x 2 = 8 choices. In general, for a set with n elements, there are 2 x 2 x ... x 2 (n times) = 2^n choices.

Therefore, the power set of a set with n elements has 2^n elements.

So, in our case, if a is a finite set with |a| elements, then the power set of a (s) has 2^|a| elements.

Therefore, |s| = 2^|a|, which proves that |s| = 2|a|.

I hope that helps! Let me know if you have any other questions.
Hi! The given statement is true. If 'a' is a finite set, then 's' is the set of all subsets of 'a', also known as the power set of 'a'. The cardinality (or size) of the power set, denoted by |s|, is equal to 2 raised to the power of the size of the finite set 'a', i.e., |s| = 2^|a|.

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The solution to the initial value problem. y' (t) - 2y = 6, y(2) = 2 is  y(t) = -3 + (5/e^4)e^(2t)..

To solve the initial value problem y'(t) - 2y = 6 with y(2) = 2,

we will use an integrating factor and the given initial condition. Here's the step-by-step solution:

1. Identify the integrating factor: The integrating factor is e^(-2t).

2. Multiply the equation by the integrating factor: e^(-2t)y'(t) - 2e^(-2t)y = 6e^(-2t).

3. Observe that the left side of the equation is the derivative of the product y(t)e^(-2t): (y(t)e^(-2t))' = 6e^(-2t).

4. Integrate both sides with respect to t: ∫(y(t)e^(-2t))' dt = ∫6e^(-2t) dt.

5. Integrate: y(t)e^(-2t) = -3e^(-2t) + C.

6. Solve for y(t): y(t) = -3 + Ce^(2t).

7. Apply the initial condition y(2) = 2: 2 = -3 + Ce^(4).

8. Solve for C: C = (5/e^4).

9. Substituting C back into the solution for y(t): y(t) = -3 + (5/e^4)e^(2t).
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the Taylor polynomial T3(x) for the function f(x) = x[tex]e^{(-5x)}[/tex]centered at a = 0 is T3(x) = 1 + 0.25[tex]x^{2}[/tex] - 0.0083[tex]x^{3}[/tex]

To find the Taylor polynomial T3(x) for the function f(x) = x[tex]e^{(-5x)}[/tex]centered at a = 0, we need to find the first four derivatives of f(x) and evaluate them at x = 0:

f(x) = x[tex]e^{(-5x)}[/tex]

f'(x) = [tex]e^{(-5x)}[/tex] - 5x[tex]e^{(-5x)}[/tex]

f''(x) = 25x[tex]e^{(-5x)}[/tex] - 20[tex]e^{(-5x)}[/tex]

f'''(x) = -125x[tex]e^{(-5x)}[/tex] + 75[tex]e^{(-5x)}[/tex]

f''''(x) = 625x[tex]e^{(-5x)}[/tex] - 500[tex]e^{(-5x)}[/tex]

Now, we can write the Taylor polynomial T3(x) as:

T3(x) = f(0) + f'(0)x + (f''(0)/2!)[tex]x^{2}[/tex] + (f'''(0)/3!)[tex]x^{3}[/tex]

T3(x) = f(0) + f'(0)x + (f''(0)/2!)[tex]x^{2}[/tex] + (f'''(0)/3!)[tex]x^{3}[/tex]

T3(x) = f(0) = 0 + f'(0)x = [tex]e^{0}[/tex] × 1 - 5 × 0 × [tex]e^{0}[/tex] = 1

T3(x) = f(0) + f'(0)x + (f''(0)/2!)[tex]x^{2}[/tex] = 1 + 0.25[tex]x^{2}[/tex]

T3(x) = f(0) + f'(0)x + (f''(0)/2!)[tex]x^{2}[/tex] + (f'''(0)/3!)[tex]x^{3}[/tex] = 1 + 0.25[tex]x^{2}[/tex] - 0.0083[tex]x^{3}[/tex]

Therefore, the Taylor polynomial T3(x) for the function f(x) = x[tex]e^{(-5x)}[/tex]centered at a = 0 is T3(x) = 1 + 0.25[tex]x^{2}[/tex] - 0.0083[tex]x^{3}[/tex]

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The correct answer is 26%.

In an exponential regression model, the exact percentage of change can be calculated as: (exp(β1) - 1) × 100. Given that β1 = 0.23, let's calculate the percent increase in E(y):

Step 1: Calculate exp(β1): exp(0.23) ≈ 1.259
Step 2: Subtract 1: 1.259 - 1 = 0.259
Step 3: Multiply by 100: 0.259 × 100 = 25.9

The percent increase in E(y) is approximately 26%. So, the correct answer is 26%.

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The Correct Option is C that 71 x 10-² of the following gives the value of the expression below written in scientific notation.

The basic objective of scientific notation is to make computations with unusually big or small numbers simpler. The following examples demonstrate how all of the digits in a number in scientific notation are relevant because zeros are no longer utilised to set the decimal point.

The statement for a number n in scientific notation is of the type a10b, where an is an integer such that 1|a|10. B is also an integer. Multiplication: To get the full amount in scientific notation, multiply the decimal values. Add the 10 power exponents after that.

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The absolute value equation is:

|x - 15| = 0

Here, we have,

to write the absolute value equation:

We want an absolute value equation that only has the solution x = 15.

So we must have something equal to zero (so we avoid the problem with the signs that we can have with other numbers)

So the equation will be something like:

|x - a| = 0

And the solution is 15, so:

|15 - a | = 0

then a = 15

The equation is:

|x - 15| = 0

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Answer:

|x - 15| = 0

Notes:

You're probably going to give the other person Brainliest, but could you maybe consider giving it to me?

If a card is drawn from a "standard-deck" and then month of year is chosen, then the probability of selecting "face-card" and June month is 1/52.

The probability of getting a face-card from a standard-deck of 52 cards is 12/52, since there are 12 face cards (four jacks, four queens, and four kings) in the deck.

The probability of choosing June from the 12 months of the year is 1/12, since there are 12 months in a year and each month is equally likely to be chosen.

To find the probability of both events happening together (getting a face-card and June), we multiply the probabilities of each event:

P(face card and June) = P(face card) × P(June) = (12/52) × (1/12) = 1/52

Therefore, the probability of getting a face card and June is 1/52.

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The probabilities of

a) P(3) = 0.07

b) P(<1) = 0.67

c) P(≤2) = 0.93

d) P(>2) = 0.07

Probability is a measure of the likelihood of an event occurring. It is a numerical value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

a) The probability of getting a value of 3 is simply the value of P(3),

P(3) = 0.07

b) To find the probability of getting a value less than 1, we add the probabilities of getting 0 and 1,

P( < 1) = P(0) + P(1) = 0.02 + 0.65 = 0.67

c) To find the probability of getting a value less than or equal to 2, we add the probabilities of getting 0, 1, and 2,

P( ≤ 2) = P(0) + P(1) + P(2) = 0.02 + 0.65 + 0.26 = 0.93

d) To find the probability of getting a value greater than 2, we simply look at the probability of getting a value of 3, which is 0.07.

P( > 2) = P(3) = 0.07

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The given question is incomplete, the complete question is:

X 0 1 2 3

P(x) .02 .65 .26 .07

Find the probabilities

a) P(3) =

b) P( < 1) =

c) P( ≤ 2) =

d) P (>2) =

Therefore, [tex]P(X^2 \leq x, Y^2 \leq y) = P(X^2 \leq x) P(Y^2 \leq y)[/tex] for all x, y in [0, ∞), which shows that [tex]X^2[/tex] is independent of [tex]Y^2.[/tex]

Here X and Y be non-negative independent random variables. We want to prove that  [tex]X^2[/tex] is independent of [tex]Y^2.[/tex]

Using the definition of independence, we need to show that the joint distribution of  [tex]X^2[/tex] is independent of [tex]Y^2.[/tex] can be expressed as the product of their marginal distributions. That is,

[tex]P(X^2 \leq x, Y^2 \leq y) = P(X^2 \leq x) P(Y^2 \leq y)[/tex]

for all x, y in [0, ∞).

We have

[tex]P(X^2 \leq x, Y^2 \leq y) = P(X \leq \sqrt{x}, Y \leq \sqrt{y}) = P(X \leq \sqrt{x}) P(Y \leq \sqrt{y})[/tex]

by the independence of X and Y.

Now, using the fact that X and Y are non-negative, we have

[tex]P(X^2 \leq x) = P(-\sqrt{x} \leq X \leq \sqrt{x}) = P(X \leq \sqrt{x}) - P(X < -\sqrt{x}) = P(X \leq \sqrt{x})\\ P(X < -\sqrt{x}) = 0.[/tex]

Similarly, we have [tex]P(Y^2 \leq y) = P(Y \leq \sqrt{y}).[/tex]

Therefore, [tex]P(X^2 \leq x, Y^2 \leq y) = P(X^2 \leq x) P(Y^2 \leq y)[/tex]

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A general solution

[tex]y(x) = c1e^{3x} + c2e^{-x} + c3*e^{-4x}[/tex]

To verify that the given functions form a fundamental solution set for the differential equation y''' + 2y" - 11y' - 12y = 0, we can use the Wronskian. The Wronskian is defined as:

W(x) = | y1(x) y2(x) y3(x) |

| y1'(x) y2'(x) y3'(x) |

| y1''(x) y2''(x) y3''(x) |

where y1(x), y2(x), and y3(x) are the given functions.

Using the given functions, we can compute the Wronskian as follows:

W(x) = |[tex]e^{3x} e^{-x} e^{-4x} || 3e^{3x} -e^{-x} -4e^{-4x} || 9e^{3x} e^{-x} 16e^{-4x}[/tex]|

Expanding the determinant, we get:

[tex]W(x) = e^{3x}(-e^{-x}*16e^{-4x} + e^{-4x}e^{-x}) - (-e^{-x}(-4e^{-4x}) - (-e^{3x})*16e^{-4x})e^{3x} + (3e^{3x}(-e^{-x}*e^{-4x}) - e^{-x}*9e^{3x}*16e^{-4x})[/tex]

Simplifying, we get:

W(x) = -23e^(-3x)

Since the Wronskian is nonzero everywhere, the functions {e^(3x), e^(-x), e^(-4x)} form a fundamental solution set for the differential equation.

To find the general solution of the differential equation, we can use the formula:

y(x) = c1y1(x) + c2y2(x) + c3*y3(x)

where c1, c2, and c3 are constants. Substituting the given functions, we get:

[tex]y(x) = c1e^{3x} + c2e^{-x} + c3*e^{-4x}[/tex]

This is the general solution of the given differential equation.

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To write out the joint probability for the given sentence using the chain rule, we can express it as follows: p(There, is, only, one, person, who, is, not, ordinary) =

p(There) * p(is | There) * p(only | is, There) * p(one | only, is, There) *, p(person | one, only, is, There) * p(who | person, one, only, is, There) *,p(is | who, person, one, only, is, There) * p(not | is, who, person, one, only, is, There) *,p(ordinary | not, is, who, person, one, only, is, There).

This expression applies the chain rule to break down the joint probability of the sentence into the product of the conditional probabilities of each word given the preceding words in the sentence. To write out the probability using the second-order Markov assumption, we assume that the probability of each word only depends on the two preceding words. Therefore, we can express the probability as follows:

p(There, is, only, one, person, who, is, not, ordinary) = p(There) * p(is | There) * p(only | There, is) * p(one | is, only) *
p(person | only, one) * p(who | one, person) * p(is | person, who) *
p(not | who, is) * p(ordinary | is, not)
This expression only considers the two preceding words for each word in the sentence, which is the second-order Markov assumption.

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