Assume that we have two events, A and B, that are mutually exclusive. Assume further that we know P(A) = 0.30 and P(B) =0.40.
1. What is P(A B)?
2. What is P(A | B)?
3. Is P(A | B) equal to P(A)?
4. A student in statistics argues that the concepts of mutually exclusive events and independent events are really the same, and that if events are mutually exclusive they must be independent. Is this statement accurate?
5. What general conclusion would you make about mutually exclusive and independent events given the results of this problem?

The equation AB = BC for A is A = BC[tex]B^{-1}[/tex].

As per the given data:

The equation AB = BC for A is given.

Also, we have to assume that A, B, and C are square matrices and that B is invertible.

Here we have to solve the equation AB = BC for A.

If the determinant of matrix A is non-zero, then the inverse of that matrix exists.

The inverse of matrix A is denoted by [tex]$A^{-1}$[/tex]. If a matrix is invertible, then the condition [tex]$A A^{-1}=I=A^{-1} A$[/tex] will be true.

Any matrix A multiplied by the identity matrix I will produce the same matrix A as the output, i.e., AI = IA = A.

Since B is invertible there exists [tex]B^{-1}[/tex]

[tex]\Rightarrow A B B^{-1}=B C B^{-1} \quad\left[/tex]

Solve the obtained equation for A by using the properties B[tex]B^{-1}[/tex]

[tex]$\Rightarrow A I=B C B^{-1}$[/tex] where I is the identity matrix of same order.

A = BC[tex]B^{-1}[/tex] because A I = I A = A

Therefore, the required matrix A = BC[tex]B^{-1}[/tex] .

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Answer:

32

Step-by-step explanation:

Answer:

The answer is D

Step-by-step explanation:

because u need to divide the y and x and when u do it to all of them u will get the same ratio which is 4

The x-component of the total force exerted on the third charge by the other two charges is equal to k * (30.00 nC * 4.00 cm / 5.00 cm + 12.00 nC).

Force is a physical quantity that measures the influence of one object on another. It is a vector quantity, meaning it has both magnitude and direction.

Let's use Coulomb's Law to calculate the x-component of the total force exerted on the third charge by the other two charges. The x-component of the total force is just the sum of the x-components of the forces exerted by each of the two charges.

First, we'll calculate the force exerted by the 5.00 nC charge at the origin. The direction of this force will be towards the third charge, so the angle θ is the arctangent of the ratio of the y-coordinate of the third charge and the x-coordinate of the third charge. That is, θ = arctan(3.00 cm / 4.00 cm). Using Coulomb's Law, we have:

F = k * q1 * q2 / r²,

where k is the Coulomb constant, q1 and q2 are the magnitudes of the charges, and r is the distance between them. For this case,

=> r = √(4.00 cm² + 3.00 cm²) = 5.00 cm.

Thus,

F = k * 5.00 nC * 6.00 nC / 5.00 cm^2 = k * 30.00 nC.

Fx = F * cos(θ) = k * 30.00 nC * cos(arctan(3.00 cm / 4.00 cm))

Fx = k * 30.00 nC * 4.00 cm / 5.00 cm.

Next, we'll calculate the force exerted by the -2.00 nC charge on the x-axis. This force will be in the negative x-direction, so the angle θ is 180 degrees. Thus,

F = k * -2.00 nC * 6.00 nC / 4.00 cm^2 = -k * 12.00 nC.

Fx = F * cos(180) = -k * 12.00 nC * cos(180) = k * 12.00 nC.

Finally, we add up the x-components of the forces exerted by each charge to get the total x-component of the force:

Fx = k * 30.00 nC * 4.00 cm / 5.00 cm + k * 12.00 nC

Fx = k * (30.00 nC * 4.00 cm / 5.00 cm + 12.00 nC).

Complete Question:

A charge of 5.00 nC is placed at the origin of the xy - coordinate system, and a charge of -2.00 nC is placed on the positive x-axis at x = 4.00 cm. A third particle, of charge 6.00 nC is now placed at the point x = 4.00 cm, y = 3.00 cm.

Find the x-component of the total force exerted on the third charge by the other two.

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The linear function through the points (-8,-2) and (-4,6) is given as follows:

y = 2x + 14.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

Given two points, the slope is calculated as the change in y divided by the change in x, hence:

m = (6 - (-2))/(-4 - (-8))

m = 8/4

m = 2.

Hence:

y = 2x + b.

When x = -8, y = -2, hence the intercept b is obtained as follows:

-2 = 2(-8) + b

b = 14.

Hence the equation is:

y = 2x + 14.

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Answer:

84 units

Step-by-step explanation:

The darkest shaded face has 12 full units. Times that by the width or the lightest shade area (7) you get the answer 84 units.

12(7) = 84

The value of the given expression will be 4.

Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.

Given that the expression is 2⁴ / ( 3.2 x 0.8 0 + 1.44 ). SOlve the expression as below,

E = 2⁴ / ( 3.2 x 0.8 ) + 1.44 )

Solve the denominator by mathematical operation.

E = 16 / ( 2.56 + 1.44 )

Divide the numerator by denominator,

E = 16 / 4

E = 4

Therefore, the solution of the expression is 4.

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Answer:

Step-by-step explanation:

Ans:percentage increased by 20

Gross profit/Cost of sales=20percent

Gross profit=20percent x cost of sales

Sales=120percent x cost of sales

Sales=120x40

        =$48

48-40=$8

8/40x100percent=20percent

Answer: Decrease? not 100% sure

Step-by-step explanation:

It takes 2 years for the account balance to come down to 0.

Compound interest is the interest on savings calculated on both the initial principal and the accumulated interest from previous periods.

The formula used to find the compound interest = [tex]A=P(1+\frac{r}{100})^{nt}[/tex].

Given that, principal =$20,000, rate of interest =1.56%, n=52 and W=$200.

Use the formula, [tex]P=W\frac{1-(1+\frac{r}{n})^{-nt} }{\frac{r}{n} }[/tex]

[tex]20000=200\frac{1-(1+\frac{0.0156}{52})^{-52t} }{\frac{0.0156}{52} }[/tex]

[tex]100=\frac{1-(1+0.0003)^{-52t}}{0.0003}[/tex]

[tex]0.03=1-(1.0003)^{-52t}[/tex]

[tex]0.97=(1.0003)^{-52t}[/tex]

Take log on both the side, we get

-52t·log(1.0003)=log0.97

So, t=2 years

Therefore, it takes 2 years for the account balance to come down to 0.

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Answer: put tow points, 1  on the x-axis (horizontal) and the other 1 on y-axis (vertical).

x = 7

y = -8

and you make a straight line , connecting them

(h, k+1/4a) are the coordinates for the focus of parabola. And the standard form is (x - h)2 = 4p (y - k).

To get the focus of a parabola, you must be aware that the equation of a parabola in a vertex form is y=a(xh)2+k, where a denotes the slope of the equation. The center of the parabola is at (h, k+1/4a), according to the formula. The focal point of the parabola is its axis. The centre of a parabola is 'a' units away from the vertex. The vertex and the focus are positioned on the parabola's axis, which you may identify using the equation for the parabola.

The axis of the parabola with a second-degree x equation is the y axis, or a line parallel to it, whereas the axis of the parabola with a second-degree y equation is the x axis, or a line parallel to it.

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The score on the seventh test is 71.5

"Information available from the question"

The average of your first 6 tests is 82.

If your test average after your seventh test is an 80. 5

To find the score on the seventh test.

Now, According to the information:

Average of 6 test is 82

We know that :

Average = Sum of all test / Total no. of test => 82 =  Sum of all test / 6

Sum of all test = 492

Let the seventh test be 'x'

So, Sum of seventh test is = 492 + x

Average = [tex]\frac{492+x}{7}[/tex]

80.5  = [tex]\frac{492+x}{7}[/tex]

=> 80.5 × 7 = 492 + x

x =  (80.5 × 7) - 492

=> x = 563.5 - 492

x = 71.5

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A good residual plot should have the points randomly scattered around the horizontal line with no discernible pattern.

This indicates that the model is well fit to the data. The residuals should also be equally distributed on either side of the horizontal line. This can be seen as the sum of the residuals being equal to 0, or mathematically expressed as the sum of the residuals (e) being equal to 0, where [tex]e = y - ŷ[/tex], where y is the observed value and ŷ is the predicted value. The equation can be expressed as Σe=0.

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The 5 most common Pythagorean triples are ( 3 , 4 , 5 ) , ( 5 , 12 , 13 ) ,

( 6 , 8 , 10 ) , ( 9 , 12 , 15 ) , and ( 15 , 20 , 25 ) .

Pythagorean triples:

Any three positive integers that completely satisfy the Pythagorean theorem is considered a Pythagorean triple number. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a right triangle. A Pythagorean triple consists of the three sides of a right triangle. In this article, you will learn how to create multiple Pythagorean triples.

A Pythagorean triple consists of the three natural numbers a, b, c such that a² + b² = c². These triples are usually written as (a, b, c), examples

(3, 4, 5) are well known. If (a, b, c) is a Pythagorean triple, then (ka, kb, kc) is also a Pythagorean triple for all natural k. A primitive Pythagorean triplet is a triplet in which a, b, and c are coprime (i.e., have no common divisor greater than 1). For example, (3, 4, 5) is a basic Pythagorean triple, but (6, 8, 10) is not.

A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle and is necessarily a right triangle.

This name comes from the Pythagorean theorem, which states that the side lengths of each right triangle satisfy the formula a² + b² =c². A Pythagorean triple gives the length of three integers of the sides of a right triangle. However, right triangles whose sides are not integers do not form a Pythagorean triple. For example, a triangle with sides a = b = 1 and c = √2 but, (1,1,2) is not a Pythagorean triple because √2 is not an integer. Also, the numbers 1 and √2 do not have a common multiple because √2 is irrational.

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The graph is a linear function with a growth factor of 4

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

The graph has the shape of a straight line, so it is linear.

The graph starts at point (0, 0). It passes through point (1, 4).

slope = 4/1 = 4

The slope is 4. Growth factor is the same as slope for a linear function.

Hence, the graph is a linear function with a growth factor of 4

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Answer: -11

Step-by-step explanation:

The evaluation of the expressions are shown below

To determine the solution to the expressions, we make use of laws of indices and exponents

Expression 1 (a) and (b)

(27)^(-4/3) ÷ (1/64)^(2/3) = (3^3)^(-4/3)  ÷ (1/4^3)^(2/3)

(27)^(-4/3) = (3)^(-4)  ÷ (1/4)^(2)

(27)^(-4/3) = 1/81 ÷ 1/16

So, we have

(27)^(-4/3) = 16/81

Next, we have

(32)^(2/5) - (-125)^(-1/3) = (2^5)^(2/5) - (-5^3)^(-1/3)

(32)^(2/5) - (-125)^(-1/3) = (2)^(2) - (-5)^(-1)

(32)^(2/5) - (-125)^(-1/3) = 4 + 1/5

(32)^(2/5) - (-125)^(-1/3) = 21/5

Equation (2)

Here, we have

3(8/27)^(-2/3) = 2^x/3^y

So, we have

3(3/2)^(2) = 2^x/3^y

So, we have

3^3/2^2 = 2^x/3^y

By comparison, we have

x = -2 and y = 3

Equation (3)

Here, we have

3^x(9^[1 - x]) = 27^[x + 2]

So, we have

3^x(3^2[1 - x]) = 27^[x + 2]

This gives

3^(x + 2[1 - x]) = 3^3[x + 2]

By comparison, we have

x + 2[1 - x] = 3[x + 2]

So, we have

x + 2 - 2x = 3x + 6

Evaluate

4x = -4

x = -1

Equation (4)

Here, we have

(5^x)^2x = 25(1/125)^x

So, we have

(5)^2x^2 = 5^[2 - 3x]

By comparison, we have

2x^2 = 2 - 3x

So, we have

2x^2 + 3x - 2 = 0

Evaluate

x = -2 and x = 1/2

Equation (5)

Here, we have

2^x+1 + 2^3-x = 10

Here, we make use of a graphing tool

So, we have

x = 0 and x = 2

Expression (6)

Here, we have

√16^x+2

Express 16 as 2^4

So, we have

√16^x+2 = √2^4[x+2]

Take the square roots

√16^x+2 = 2^2[x+2]

Open the bracket

√16^x+2 = 2^2x+4

Hence, the expression is 2^2x+4

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Here we have a program written in C++ language. Each // symbol is a relative comment explaining the reason for each command line (and is not included during the program execution).

The Program:

#include <iostream>

//c++ build in library

using namespace std;

//main code body starts here

int main()

{

 //declare variable to store phone numbers,its area code, prefix and line    number.

long pnumber;   //declare long variable

int ac, prefix, lnumber;

 //declare integer variables, where ac: Area Code and lnumber is the Line Number

cout<<"Enter a 10-digit Phone Number: "<<endl;

 //cout command prints on screen the desired message

 cin>>pnumber;

 //cin command enables the user to interact with the programm and enter information manually

//main body to obtain the desired output starts below

 //each 'division' is used to allocate the correct value at the correct

 //since prefix is used to get the desired output of ( )    -

 ac = pnumber/10000000;

 prefix = (pnumber/10000)%1000;

 lnumber = pnumber%10000;

 //main body ends here

 cout<<"Based on your 10-digit number"<<endl;

 cout<<"below you have (AreaCode), Prefix, and - line number "<<endl;

 cout<<"("<<ac<<")"<<""<<prefix<<"-"<<lnumber<<endl;

 //Prints on screen the desired output of the long 10 digit Number

 return 0; //ends program

}

-------------------------------------------------------------------------------------------------------

The Output Sample:

Enter a 10-digit Phone Number:

8019004673

Based on your 10-digit number

below you have (Area Code), Prefix, and - line number

(801) 900-4673

....Programm finished with exit code 0

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Answer:

7.40

Step-by-step explanation:

51.8 divided by 7 is 7.4

The base of the triangle with area 14ft² and height 6ft is 4⅔ feet

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted by triangle ABC.

The area of a triangle is given by ;

1/2bh

Area = 14ft²

height = 6ft²

therefore;

14 = 1/2 b × 6

28 = 6b

divide both sides by 6

b = 28/6

b = 14/3

b = 4⅔ ft

therefore the base of the triangle is 4⅔ feet.

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Answer:

You probably know that it is a good idea to brush your teeth every day. What does this do that helps prevent tooth decay?

Step-by-step explanation

A positive divided by a negative always results in a negative quotient.

Let's use the example 8 / (-2) again. We can write this expression as:

8 / (-2) = 8 / (-1) * 2

We can then simplify the expression on the right-hand side by multiplying -1 and 2, which gives us:

8 / (-1) * 2 = -8 / 2

Now, we can divide 8 by 2, which gives us:

-8 / 2 = -4

So we see that the quotient is negative.

We can generalize this to any positive number divided by a negative number. Let's say we have a positive number a and a negative number -b. We can write the expression as:

a / (-b) = a / (-1) * b

Multiplying -1 and b gives us:

a / (-1) * b = -a / b

So we see that the quotient is negative, since -1 times b is negative, and dividing a positive number by a negative number results in a negative quotient.

Therefore, we can conclude that when a positive number is divided by a negative number, the result is always negative.

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Answer: 0.429

Step-by-step explanation: Use a calculator and put the fraction and hit enter to get it in decimal form. You should get a number like 0.42857142857 but the problem asks us to round to the nearest thousandth. Since there is a 5 in the ten-thousandth place, we must round up. Therefore, the final answer will be 0.429.

Explanation:

The missing angle between the 107 and (4-2x) is 55 due to the vertical angles theorem.

The three angles mentioned add to 180 because they form a straight line.

Here's what the steps would look like:

107+55+(4-2x) = 180

166-2x = 180

-2x = 180-166

-2x = 14

x = 14/(-2)

x = -7

The perimeter of the polygon with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2) is 16 units.

Perimeter is a circumferential measure of the figure, for example, the perimeter of a square is the sum of all its sides.

Here,

To find the perimeter of a polygon, we need to add up the lengths of all its sides. In this case, we have a quadrilateral with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2).

We can use the distance formula to find the lengths of each side:

Length of QR = √[(1 - (-3))² + (2 - 2)²] =  4

Similarly,

Length of RS= 4

Length of ST= 4

Length of TQ= 4

Therefore, the perimeter of the polygon is 4 + 4 + 4 + 4 = 16

So the perimeter of the polygon with vertices Q(-3, 2), R(1, 2), S(1,-2), and T(-3,-2) is 16 units.

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The slope of line from the points given is 1

The slope of a line is a measure of how steep the line is and the direction it is going. In mathematical terms, it is defined as the ratio of the change in the y-coordinate (rise) to the change in the x-coordinate (run) between two points on the line. Slope is often represented by the letter "m". If (x₁, y₁) and (x₂, y₂) are two points on the line, the slope can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

In this problem, we can take the two points and calculate the slope.

m =  (y₂ - y₁) / (x₂ - x₁)

Substituting the values into the formula to find the slope;

m = -5 - 1 / -8 - (-2)

m = 1

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Complete Question; Calculate the slope of the line (-2, 1) and (-8, -5)

Note that the proof required is given based on the AA Similarity Postulate.

Similar triangles: If two triangles are similar then their corresponding angles are equal.

By the transitive property of equality if a = b, and b= c then a = c.

AA postulate of similarity states that when two corresponding angles of two triangles are equal then they are called similar to each other.

Given ΔABC ~ ΔRST; and

ΔDEF ~ ΔRST

To Prove: ΔABC ~ DEF Note that:

ΔABC ~ ΔRST - Given

ΔDEF ~ ΔRST
- Given

∠A  = ∠R, ∠D = ∠R - Definition of Similar Triangles

∠C = ∠T, ∠F = ∠T

∠A= ∠C, ∠D = ∠F  - Transitive property of Equality; Thus,

ΔABC ~ Δ DEF (AA Similarity Postulate)

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Answer: No, Alan's statement is incorrect!

Step-by-step explanation:

The intersection point of a system of inequalities is not always a solution. It only becomes a solution if the two lines meet at the same point and the lines are not parallel. If the two lines are parallel, then they will not intersect and thus the intersection point will not be a solution.

Step-by-step explanation:

if there is an intersection point (the lines are not parallel), there is still the question, if the points on the line are included in the solution or not.

if not, then the intersection point is not part of the solution either.

this is controlled by the absence of presence of the "or equal" part of the inequality.

>= greater than or equal

<= less than or equal

if the inequality is only

> greater than

< less than

the lines are just delimiters, but their points are not part of the solution. and therefore not the intersection point either.

The expression in simplest form that represents the total cost for x students is 11.55x, the correct option is C.

Expression in mathematics can be defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

Given that a student in a math class has to buy the items in the table.

Now,

x(9.65+0.65+1.25)

=9.65x+0.65x+1.25x

=11.55x

Therefore, the expression in simplest form will be 11.55x.

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Answer:

X = 6/ y+3 - 2

and

Y -3x/x+2

Step-by-step explanation:

hope this helps