In a class of students, the following data table summarizes how many students passed
a test and complete the homework due the day of the test. What is the probability that
a student completed the homework given that they failed the test?
Completed the homework
Did not complete the homework
Passed the test Failed the test
8
2
3
сл

The equation of the quadratic function represented by the given table is  y = -x² + 4x - 7.

A quadratic function is a function of the form:\sf(x) = ax^2 + bx + c\swhere a, b, and c are constants and x is the parameter. The graph of a quadratic function is a parabola, which is an Inverted curve. Whether the parabola opens up (if a > 0) or down (if a 0) depends on the sign of the coefficient a.

The width of the parabola is also determined by the coefficient a. The parabola is narrow if |a| is greater than 1. (i.e. it has a small width relative to its height). The parabola is wide if |a| is greater than 1.

The standard form of the quadratic equation is given as:

y = ax² + bx + c

Substitute the value of x and y from the table:

3 = a(2)² + b(2) + c

4a + 2b + c = 3........(1)

For point (4, -1):

-1 = a(4)² + b(4) + c

16a + 4b + c = -1..........(2)

For (6, -13):

-13 = a(6)² + b(6) + c

36a + 6b + c = -13..........(3)

From 1 we have:

c = 3 - 4a - 2b

Substitute the value of c in equation 2 and 3:

16a + 4b + 3 - 4a - 2b = - 1

12a + 2b = - 4........(4)

36a + 6b + 3 - 4a - 2b = -13

32a + 4b = -16.......(5)

Multiply equation 4 with 2 and subtract with equation 5:

32a + 4b = -16

-(24a + 4b = - 8)

a = -1

Substitute the value of a in equation 5:

32(-1) + 4b = -16

-32 + 4b = -16

b = 4

Substitute the value of a and b in equation 1:

16a + 4b + c = -1

16(-1) + 4(4) + c = -1

-16 + 8 + c = -1

-8 + c = -1

c = 7

Using the algebraic techniques we have:

a = -1

b = 4

c = 7

Hence, the equation of the quadratic function represented by the given table is y = -x² + 4x - 7.

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From the given information provided, the value of angle LMP and angle NMP is 77 and 103 degrees respectively.

Since LMN is a straight angle, it measures 180 degrees.

We are given the measures of LMP and NMP, and we are told that LMP + NMP = LMN. Therefore, we can set up an equation:

LMP + NMP = LMN

(-16x + 13) + (-20x + 23) = 180

Simplifying and solving for x, we get:

-36x + 36 = 180

-36x = 144

x = -4

Now that we have found the value of x, we can substitute it back into the expressions for LMP and NMP to find their measures:

LMP = -16x + 13 = -16(-4) + 13 = 77 degrees

NMP = -20x + 23 = -20(-4) + 23 = 103 degrees

Therefore, the measures of LMP and NMP are 77 degrees and 103 degrees, respectively, and the measure of LMN is 180 degrees.

Question - LMN is a straight angle. LMP = -16x + 13 NMP =  -20x + 23 LMP + NMP = LMN What are the measures?

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The growth rate of a bacterial colony after a 2 hours  is given by the derivative of its population function with respect to time is 36 .

The growth rate of a bacterial colony is given by the derivative of its population function.

Thus, we need to find the derivative of the population function p(t) with respect to time t, and then evaluate it at t = 2 to get the growth rate after 2 hours.

p(t) = 3t² + 24t + 200

Taking the derivative of p(t) with respect to t, we get:

p'(t) = 6t + 24

Now, evaluating p'(t) at t = 2, we get:

p'(2) = 6(2) + 24 = 36

Therefore, the growth rate of the bacterial colony after 2 hours is 36.

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D: "[tex]c_{1} = 10[/tex] and [tex]c_{2} = 2[/tex]" are programmer-defined constants for quadratic probing that cannot be used in a quadratic probing equation. Option D is correct answer.

The quadratic probing equation is defined as:

h (k, i) = (h′(k) + [tex]c_{1}[/tex] * i + [tex]c_{2}[/tex] * i^2) mod m,

where h′(k) is the hash value of key

k and m is the size of the hash table.

The constants [tex]c_{1}[/tex] and [tex]c_{2}[/tex] are programmer-defined constants that are used to compute the new hash index when a collision occurs in the hash table.

The given hash function is h(k) = k % 10.

Therefore, the hash value of any key will be between `0` and `9`.Now, let's check which of the given programmer-defined constants for quadratic probing cannot be used in a quadratic probing equation:

Option A: `c1 = 1 and c2 = 0`This option can be used in the quadratic probing equation. It means that linear probing is being used.

Option B: [tex]c_1 = 5[/tex] and [tex]c_2 = 1[/tex] This option can be used in the quadratic probing equation. It means that the new index is being computed as `h(k, i) = (h′(k) + 5i + i^2) mod m`.

Option C: [tex]c_1 = 1[/tex] and [tex]c_2 = 5[/tex] This option can be used in the quadratic probing equation. It means that the new index is being computed as `h(k, i) = (h′(k) + i + 5i^2) mod m`.

Option D: [tex]c_1 = 10[/tex] and [tex]c_2 = 2[/tex] This option cannot be used in the quadratic probing equation. It means that the new index is being computed as `h(k, i) = (h′(k) + 10i + 2i^2) mod m`.

Since [tex]c_{1}[/tex] is greater than or equal to `m`, this equation will always result in a hash index that is greater than or equal to `m`. Therefore, it is not possible to use `[tex]c_{1}[/tex]= 10` in the quadratic probing equation. Hence, the correct option is D.

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The tangential component of the acceleration vector at point t = 1 is aT(1) = 233/3 and The normal component of the acceleration vector at point t = 1 is aN(1) = (1/3)√10459

The acceleration vector can be found from the following formula:

[tex]a(t) = r''(t) = (-3,-10t,-8t3).[/tex]

To find the tangential component of the acceleration vector, we first need the velocity vector v(t).

[tex]v(t) = r'(t) = (-3,-10t,-8t3) .[/tex]

Next, we need to normalize the velocity vector using the following formula:

[tex]T(t) = v(t) / ||v(t)||,[/tex]

Where ||v(t)|| is the magnitude of the velocity vector.

[tex](1) = (-3,-10,-8) / \sqrt{(3^2 + 10^2 + 8^2)} = (-3/3, -10/3, -8/3) = (-1 , -10/3, -8/3) .[/tex]

Then, the tangential component of a(1) is:

[tex]aT(1) = a(1) T(1) = (-3, -10, -8) (-1, -10/3, -8/3) = 3 + 100/3 + 64/3 = 233/3.[/tex]

To find the normal component of a(1), we simply need to find the magnitude of the tangential component and subtract it from the magnitude of the acceleration vector.

[tex]aN(1) = \sqrt{ (a^2 - aT(1)^2)} = \sqrt{(3^2 + (10)^2 + (8)^2 - (233/3)^ 2)}  = \sqrt{(9 + 100 + 64 - 54289/9)} = \sqrt{(10459/9)} = (1/3)\sqrt{10459}[/tex]

Therefore, the tangential and normal components of the acceleration vector at the point t = 1 are:

[tex]aT(1) = 233/3[/tex] and [tex]aN(1) = (1/3)\sqrt{10459}[/tex]

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Answer:

The square root of 2,100 is approximately 45.8257569495584 when rounded to 15 decimal places.

Answer:

  D.  87

Step-by-step explanation:

You want to know the measure of the angle opposite the longest side in a triangle with side lengths 13, 16, and 20 inches.

The angle x is opposite the side of length 20 inches in this triangle, which is the longest side. That tells you x is the largest angle.

The largest angle in any triangle is never less than 60°. This eliminates all answer choices except the last one:

  x = 87

If you want to go to the trouble to solve the triangle, the law of cosines is helpful. For sides a, b, c and angle C, it tells you ...

  c² = a² +b² -2ab·cos(C)

Solving for the angle, we have ...

  C = arccos((a² +b² -c²)/(2ab))

  C = arccos((13² +16² -20²)/(2·13·16)) = arccos(25/416) ≈ 86.55°

  x ≈ 87

Answer:

Correct option is C)

Simple interest =

100

3000×6×2

=360

Compound interest =3000(1+

100

6

)

2

−3000=18×20.6=370.8

∴ Difference is Rs.10.8.

you can convert this value to $$

or simply the answer will be 2. $10

(hob-evzw-zjw) come

Answer:

B is your answer.
10.80$ which you just round to 10. 10 is your answer.

Step-by-step explanation:

For simple interest, the formula is:

Simple Interest = Principal × Rate × Time

For compound interest, the formula is:

Compound Interest = Principal × (1 + Rate)^Time - Principal

Let's calculate the values:

Principal = $3,000

Rate = 6% or 0.06

Time = 2 years

Simple Interest = $3,000 × 0.06 × 2 = $360

To calculate compound interest, we need to use the formula:

Compound Interest = $3,000 × (1 + 0.06)^2 - $3,000

= $3,000 × (1.06)^2 - $3,000

= $3,000 × 1.1236 - $3,000

= $3,370.80 - $3,000

= $370.80

The difference between simple and compound interest is:

$370.80 - $360 = $10.80

Answer:

5:3 = 5/3

Step-by-step explanation:

Given Sin∅ = 3/5

We know sin∅ = Perpendicular/Hypotenuse

And, By inverse relationship, we get

cosec∅ = Hypotenuse/Perpendicular = 1/sin∅

So, csc∅ = 5/3, 5:3

The `det(E2) of the given matrix is equal to 366`.

Given a 5 by 5 matrix E= `[11 2 -1 -3 4;10 -1 -2 -1 -2;-1 2 3 2 1;1 1 1 -1 -1;2 -1 -2 1 1]`.

To find `det(E)`, we can use the following steps.

Step 1: Create a 5 by 5 matrix E1 by adding a multiple of the ith row to the jth row, given i = 3 and j = 5.

We need to add -1/3 times the 3rd row to the 5th row. It can be done by the following operation.`E1 = E` (start with the original matrix) `=> E1(5,:) = E(5,:) - E(3,:) / 3` (subtract the 3rd row of E divided by 3 from the 5th row of E)

This results in the matrix `E1 = [11 2 -1 -3 4;10 -1 -2 -1 -2;-1 2 3 2 1;1 1 1 -1 -1;1/3 -7/3 -7/3 7/3 4/3]

`Step 2: Create a 5 by 5 matrix E2 by adding a multiple of the ith row to the jth row, given i = 2 and j = 5.We need to add -20 times the 2nd row to the 5th row.

It can be done by the following operation.`E2 = E1` (start with the matrix from Step 1) `=> E2(5,:) = E1(5,:) - 20 * E1(2,:)` (subtract 20 times the 2nd row of E1 from the 5th row of E1)

This results in the matrix `E2 = [11 2 -1 -3 4;10 -1 -2 -1 -2;-1 2 3 2 1;1 1 1 -1 -1;0 -13 33 -13 44]

`Step 3: Find det(E2) by using the cofactor expansion along the 5th column.`det(E2) = 0 - (-13) * A1 + 33 * A2 - (-13) * A3 + 44 * A4 - 0 * A5`where A1, A2, A3, A4, and A5 are the 2 by 2 determinants of the submatrices obtained by deleting the 5th row and the ith column, for i = 1, 2, 3, 4, and 5. We can use the following notation.

A1 = det([11 -1 -3 4;10 -2 -1 -2;-1 3 2 1;]) = 324A2 = det([11 2 -3 4;10 -1 -1 -2;-1 2 2 1;]) = -54A3 = det([11 2 -1 4;10 -1 -2 -2;-1 2 3 1;]) = -142A4 = det([11 2 -1 -3;10 -1 -2 -1;-1 2 3 2;]) = 50A5 = det([11 2 -1 -3;10 -1 -2 -1;-1 2 3 2;]) = 366.

Therefore `det(E2) = 0 - (-13) * 324 + 33 * (-54) - (-13) * (-142) + 44 * 50 - 0 * 50 = 366`.

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In the following question, among the various parts to solve on houses - 1. price = β0 + β1sqrft + β2bdrms, 2.  β2, 3. β2 + 140β1, 4. R-squared value is provided in the regression output, 5. 276.878 thousand dollars, 6. 23.122.

1. The regression equation of house price in thousands of dollars to the house size measured in square feet (sqft) and the number of bedrooms in the house (bdrms) can be written as follows: price = β0 + β1sqrft + β2bdrms Here, price refers to the house price in thousands of dollars, sqft refers to the house size measured in square feet and bdrms refers to the number of bedrooms in the house.

2. The estimated increase in price for a house with one more bedroom, holding square footage constant is equal to the coefficient of bdrms in the regression equation, which is β2.

3. The estimated increase in price for a house with an additional bedroom that is 140 square feet in size can be calculated as follows: β2 + 140β1. Comparing this to the answer in question two above, we can see that the price increase is greater when an additional 140 square feet are added to the house rather than an additional bedroom.

4. The percentage of the variation in price explained by square footage and the number of bedrooms can be found using the R-squared value. The R-squared value is a measure of how much of the variation in the dependent variable (house price) is explained by the independent variables (sqft and bdrms). In this case, the R-squared value is provided in the regression output.

5. To find the predicted price for the first house in the sample using the model estimated above, we need to plug in the values of sqft and bdrms for the first house into the regression equation. Here, sqrft = 2,438 and bdrms = 4. Thus, the predicted price for the first house is given by: price = β0 + β1sqrft + β2bdrms = -14.973 + 0.128sqrft + 15.204bdrms = -14.973 + 0.128(2,438) + 15.204(4) = 276.878 thousand dollars.

6. The residual for the first house in the sample can be calculated as follows: Residual = Actual price - Predicted price = 300 - 276.878 = 23.122. The fact that the residual is positive suggests that the buyer overpaid for the house.

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The population of Toledo, Ohio is increasing at a rate of approximately 32,263 people per year after 10 years.

An exponential function is a mathematical function of the form f(x) = a^x, where "a" is a positive constant called the base, and "x" is a variable that can take on any real value. The base "a" is typically greater than 1, which means the function grows at an increasing rate as "x" increases.

a. The exponential function that relates the total population as a function of t is given by:

P(t) = P₀ × (1 + r)ᵗ

where P₀ is the initial population, r is the annual growth rate (as a decimal), and t is the time in years.

Using the given values, we have:

P₀ = 500,000 (given)

r = 0.05 (5% expressed as a decimal)

Thus, the exponential function is:

P(t) = 500,000 × (1 + 0.05)ᵗ

b. The rate at which the population is increasing in t years is given by the derivative of the population function with respect to time:

dP/dt = P₀ × r × (1 + r)ᵗ

Substituting the given values, we get:

dP/dt = 500,000 × 0.05 × (1 + 0.05)ᵗ

c. To determine the rate at which the population is increasing in 10 years, we simply substitute t = 10 into the expression we derived in part b:

dP/dt = 500,000 × 0.05 × (1 + 0.05)¹⁰

Using a calculator, we get:

dP/dt ≈ 32,263

Therefore, the population of Toledo, Ohio is increasing at a rate of approximately 32,263 people per year after 10 years.

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Beth receives £103.45 when she changes €120 back into pounds.

An exchange rate is the value of one currency expressed in terms of another currency. In other words, it is the rate at which one currency can be exchanged for another currency.

Pound is a unit of currency that is used in several countries, including the United Kingdom, Egypt, Lebanon, and Sudan, among others. The pound symbol is "£".

In the given question,

If the exchange rate is £1 = €1.16, this means that for every euro, Beth will get £1/€1.16.

Therefore, the number of pounds Beth receives when she changes €120 back into pounds is:

120 euros * £1/€1.16 = £103.45 (rounded to two decimal places)

So Beth receives £103.45 when she changes €120 back into pounds.

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the exponent of the product of 8.9 x 10¹² and 4.7 x 10⁻² in scientific notation is 11.

Exponential refers to a mathematical function or relationship in which a variable (such as x) is raised to a constant power (such as 2, 3, or e) to produce a result. The term "exponential" can also be used more broadly to describe any situation in which something grows or changes at an increasingly rapid rate over time, often with a compounding effect.

First, we multiply the two numbers:

(8.9 x 10¹²) x (4.7 x 10⁻²) = 41.83 x 10¹⁰

41.83 x 10¹⁰ = 4.183 x 10¹¹

Therefore, the exponent of the product of 8.9 x 10¹²and 4.7 x 10⁻² in scientific notation is 11.

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The 21 square root of 98 divided by 7 square root of 21 = 21√98 / 7√21 = 6.4807407

A square root of a number x is a number y such that y2 = x; in other words, a number y who's square and the result of multiplying the number by itself, or y ⋅ y, is x.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √where the symbol √ is called the radical sign.

Every positive number x has two square roots: √ which is positive, and -√ which is negative. The two roots can be written more concisely using the ± although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root.

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The answer of  the given question based on the finding the area of the triangle of ABC the answer is the area of triangle ABC is approximately 62.82 square cm.

A triangle is  three-sided polygon with three angles. It is  two-dimensional geometric shape, and one of  basic shapes in geometry. A triangle can be classified based on  length of its sides and  measure of its angles. The sum of the interior angles of  triangle are 180 degrees. Triangles are used in many fields, like mathematics, engineering, architecture, and art.

To find the area of triangle ABC, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

where the base is one side of the triangle and the height is the perpendicular distance from the base to the opposite vertex.

In this case, we know that AB = 11 cm and AC = 17 cm, and angle A is 45 degrees. To find the height of the triangle, we need to use trigonometry.

First, we can find the length of BC using the Law of Cosines:

BC² = AB² + AC² - 2 * AB * AC * cos(A)

BC² = 11² + 17² - 2 * 11 * 17 * cos(45)

BC² = 156 - 265.42

BC² = 109.42

BC = 10.46 cm (rounded to two decimal places)

Now we can use the sine function to find the height of the triangle:

sin(A) = height / BC

height = BC * sin(A)

height = 10.46 * sin(45)

height = 7.39 cm (rounded to two decimal places)

Finally, we can use the area formula to find the area of the triangle:

Area = (1/2) * base * height

Area = (1/2) * 17 * 7.39

Area = 62.82 square cm (rounded to two decimal places)

Therefore, the area of triangle ABC is approximately 62.82 square cm.

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Answer:

y = 2x/15 + 6

Step-by-step explanation:

3y/2 = x/5 + 9

3y = (x/5 + 9) (2)    The 2 that was dividing goes on to multiply on the other side.

3y= 2x/5 + 18

y = (2x/5 + 18) / 3   The 3 that was multiplying goes on to divide on the other side.

y = 2x/15 + 6

Answer:

Step-by-step explanation:

27/9a

= 3^3/3^2 a

= 3/a

Answer: 2

Step-by-step explanation:

To graph the solution to the inequality -5 + x ≥ -3 on the number line, we first need to isolate x.

Adding 5 to both sides of the inequality, we get:

x ≥ 2

This means that any value of x greater than or equal to 2 will satisfy the inequality. To graph this solution on a number line, we draw a closed circle at the point 2 and shade all the points to the right of 2, including the point 2 itself.

The resulting graph looks like this:

     ------•-------------------------------->

           2

The shaded region on the right of 2 represents all the values of x that make the inequality true.

Answer:

the answer is 3/2

Step-by-step explanation:

Answer:

the answer si 3/2

The ball reaches a maximum height of approximately 146 feet, to the nearest foot.

Maximum Height refers to the highest point of the structure or sign as measured from the average natural ground level at the base of the supporting structure.

The ball is thrown upward with an initial velocity of 75 feet per second, implying that v0 = 75. We are also told that the ball is thrown from a height of 4 feet, implying that h0 = 4.

The function: can be used to model the ball's vertical motion.

16t2 + v0t + h0 = h(t).

Substituting v0 and h0 values yields:

h(t) = -16t^2 + 75t + 4

To determine the maximum height of the ball, we must first locate the vertex of the parabolic function h. (t). The vertex of the parabola is given by the equation y = ax2 + bx + c:

x = -b / 2a

y = c - b^2 / 4a

a = -16, b = 75, and c = 4 in this case. Substituting these values into the above formulas yields:

t = -75 / 2(-16) = 2.34 sec

h(t) = 4 - (752) / (4(-16)) 146 ft.

As a result, the ball reaches a maximum height of about 146 feet to the nearest foot.

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The exponential probability distribution is employed with a random variable that is continuous in nature.

What do you mean by exponential probability distribution ?

In the field of probability, a probability distribution refers to a mathematical function that gives the probabilities of various possible outcomes of an experiment. The exponential probability distribution is a probability distribution that models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. It is a continuous probability distribution, meaning that the random variable takes on values within a continuous range, as opposed to a discrete probability distribution, where the random variable takes on only a finite or countable set of values.

Explanation of the correct answer :
The exponential probability distribution is defined by a single parameter, [tex]\lambda[/tex] which represents the average rate of occurrence of events in the Poisson process.

The probability density function (pdf) of the exponential distribution is given by [tex]f(x) = \lambda e^{(-\lambda x)}[/tex], where x is the time between events. The cumulative distribution function (cdf) is given by [tex]F(x) = 1 - e^{-\lambda x}[/tex].

The exponential probability distribution is used in many applications, such as queuing theory, reliability theory, and finance. For example, it can be used to model the time between customer arrivals in a queue, the time between machine failures in a manufacturing process, or the time until default on a bond.

In summary, the exponential probability distribution is a continuous probability distribution that is used with a continuous random variable, specifically to model the time between events in a Poisson process.
Hence, option B is correct.

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The solution to the inequality f(x) ≤ 0 is the interval [-6, 2]. In other words, the values of x that satisfy the inequality are those that lie between -6 and 2, inclusive.

To solve the inequality f(x) ≤ 0, we need to find the values of x for which the function f(x) is less than or equal to zero.

We start by factoring the quadratic expression f(x) = x^2 + 4x - 12:

f(x) = (x + 6)(x - 2)

Setting this expression to zero, we get:

(x + 6)(x - 2) = 0

This gives us two solutions: x = -6 and x = 2.

Now, we need to determine the sign of f(x) in the intervals between these two solutions. We can use a sign chart to do this:

x f(x)

-∞ +

-6 0

2 0

+∞ +

From the sign chart, we see that f(x) is positive for x < -6 and for x > 2, and it is negative for -6 < x < 2.

To summarize, the solution to the inequality f(x) ≤ 0 for the function f(x) = x^2 + 4x - 12 is the interval [-6, 2].

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The error is the substitution of coordinates. Coordinates are ordered pairs of points that help us locate any point in a 2D plane or 3D space.

Cartesian coordinates, also known as the coordinates of a point in a 2D plane, are two integers, or occasionally a letter and a number, that identifies a specific point's precise location on a grid. This grid is referred to as a coordinate plane.

The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by

[tex]AB = \sqrt{(x_{1} , x_{2})^{2} + (y_{1} - y_{2})^{2} }[/tex]

Observe that the x-coordinate of B is subtracted from the x-coordinate of A. This goes with the y-coordinates.

Therefore, the error is the substitution of coordinates.

The correct computation is

[tex]AB = \sqrt{(6-1)^{2} + [2 - (-4)]^{2} }[/tex]

[tex]= \sqrt{5^{2} + 6^{2} }[/tex]

[tex]= \sqrt{25 + 36} \\[/tex]

[tex]= \sqrt{61}[/tex]

≈ 7.81

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The complete question is as follows:

Describe and correct the error in finding the distance between A(6, 2) and B(1, -4). AB = √[(6 - 2)² + {2 - (-4)}²] = √(4² + 5²) = √(16 + 25) = √41 ≈ 6.4.

A sampling distribution is normal only if the population is normal. This  statement is false because A sampling distribution is normal only if n≥30.

If the underlying population is normally distributed, the sampling distribution (such as the sample mean distribution, also known as the xbar distribution) is also normally distributed. Even though the population is not normally distributed, the x(bar) distribution is approximately normal if n > 30, due to the central limit theorem. Some textbooks may use values ​​above 30, but after a certain threshold the x(bar) distribution is effectively "normal".

Option B is close, but misses the normal population part. n > 30 is not necessary if we know the population is normal.

A sampling distribution is the probability distribution of a statistic obtained from a large number of samples drawn from a particular population. The sampling distribution for a given population is the frequency distribution of a range of different outcomes that can occur in the population.

In statistics, a population is the entire basin from which a statistical sample is drawn. A population can refer to an entire population of people, objects, events, hospital visits, or measurements. Thus, a population can be said to be a global observation of subjects grouped by common characteristics.

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The probability that no numbers are repeated = [tex]\frac{720}{1000}=0.72[/tex]

The probability that there are no repeated digits in a 3-digit pin number is 0.72.

Formula used:

[tex]P(n,r)=\frac{n!}{(n-r)!}\\ Probability=\frac{Number of favourable outcomes}{Total number of events in the samples pace}[/tex]

There are 10 digits (0,1,2,3,4,5,6,7,8,9) to choose from.

Therefore, the total number of possible 3-digit pin numbers with no repeated digits is

[tex]P(10,3)=\frac{10!}{(10-3)!}\\P(10,3)= \frac{10!}{7!}\\P(10,3)=720[/tex]

The total number of possible 3-digit pin numbers [tex]= 10 * 10 * 10 = 1000[/tex].

Thus, the probability that no numbers are repeated = [tex]\frac{720}{1000}=0.72[/tex]

Therefore, the probability that there are no repeated digits in a 3-digit pin number is 0.72.

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The steps for proof by mathematical induction that P(n) is true for all positive integers n, All options are true.

The steps for a proof by mathematical induction that P(n) is true for all positive integers n are as follows:

a. Verify that P(1) is true.

b. Demonstrate that the conditional statement Plk) implies Plk+1) is true for all positive integers k.

c. Verify that P(1), P(2), P(3), ..., P(k) is all true, where k is a specific large, positive integer.

d. Demonstrate that if P(k) is false, then Plk+1) is false for all positive integers k.

e. Demonstrate that P(k+1) implies Plk) is true for all integers k.

Therefore, option (a), option (b), option (c), option (d), and option (e) are the steps for proof by mathematical induction that P(n) is true for all positive integers n.

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Answer: 165

Step-by-step explanation:

55 x 3 = 165

The cumulative distributive function (CDF) Fx(3) is 0.61.

The cumulative distributive function (CDF) of a random variable X is the probability that X takes a value less than or equal to x. In this case, we are asked to find Fx(3).

Since the random variable X is given with a probability mass function, we can calculate the CDF by summing the probabilities of X being less than or equal to 3. This can be expressed as: [tex]Fx(3) = P(X<=3).[/tex]
For X = 0, P(X<=3) = 0.23.
For X = 1, P(X<=3) = 0.23 + 0.13 = 0.36.
For X = 2, P(X<=3) = 0.23 + 0.13 + 0.10 = 0.46.
For X = 3, P(X<=3) = 0.23 + 0.13 + 0.10 + 0.15 = 0.61.

Therefore, the cumulative distributive function (CDF) Fx(3) is 0.61.

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