We have to make choices every day. Some choices may affect our lives for years, like the colleges we attend.
Other decisions have short-term effects, like where we should eat lunch.
Read the options below. Which option would you choose?
A. Option 1: Receive $1,000,000 today.
B. Option 2: Receive $25,000 every day for a month (30 days).
C. Option 3: Start with 1 penny, then double it every day for a month (30 days).

There are some trigonometric functions such as tangent, cotangent, secant, and cosecant that have restricted ranges.

Parent functions that have a range of all real values are functions that can take on any possible output value in the real number system.

The functions that have a range of all real values include:

Constant function: f(x) = c, where c is any real number. Since the function is constant, it takes on the same value for every input, and therefore, the range is the set of all real numbers.

Linear function: f(x) = mx + b, where m and b are any real numbers. Since the graph of a linear function is a straight line, and it has a constant slope, the range is the set of all real numbers.

Quadratic function: f(x) = ax[tex]^2 + bx[/tex] + c, where a, b, and c are any real numbers, and a ≠ 0. Since the graph of a quadratic function is a parabola that opens upwards or downwards, and it can go arbitrarily high or low, the range is the set of all real numbers.

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The value of probability to get the first letter will be always be, 1 / 26.

Given that;

A computer randomly selects a letter from the alphabet.

Now, The probability the computer produces the first letter of your first name :

Here, the required outcome is getting the first letter of your first name.

Probability = No. of required outcomes / total no. of outcomes.

For example, The name Alex Davis has the first letter of the fist name as alphabet 'A'.

Hence, Probability = 1 / 26

Similarly, for any first name there is going to be any one alphabet from the 26 alphabets, thus the probability to get the first letter will be always be, 1 / 26.

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Answer:

3x(x + 3) = 0

Step-by-step explanation:

3x² + 9x = 0 ← factor out common factor of 3x from each term on the left

3x(x + 3) = 0 ← factored form

After approximately 60.2 minutes, the pizza will reach a temperature of 150°F.

We can model the cooling of the pizza using Newton's law of cooling:

T(t) = T_room + (T_0 - T_room) e^(-kt)

where T(t) is the temperature of the pizza at time t, T_0 is the initial temperature of the pizza (300°F), T_room is the room temperature (70°F), and k is the cooling rate constant. We can solve for k using the fact that the pizza cools from 300°F to 100°F in M minutes:

100 = 70 + (300 - 70) e^(-kM)

e^(-kM) = 0.1

-kM = ln(0.1)

k = -ln(0.1)/M

Now we want to find the time t when the pizza's temperature is 150°F:

150 = 70 + (300 - 70) e^(-kt)

e^(-kt) = (150 - 70)/(300 - 70)

e^(-kt) = 4/13

-kt = ln(4/13)

t = -ln(4/13)/k

Substituting the expression for k derived earlier, we get:

t = M ln(4/13) / ln(0.1)

For example, if M = 30 (i.e., it takes 30 minutes for the pizza to cool from 300°F to 100°F), then:

t = 30 ln(4/13) / ln(0.1) ≈ 60.2 minutes

Therefore, after approximately 60.2 minutes, the pizza will reach a temperature of 150°F.

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Phrase 1: A. is at most (5 2) = 10
Phrase 2: B. is at most (6 2) = 15
Phrase 3: E. equals 5 + 7 = 12
Phrase 4: E. is at most 11 + 13 + 12 = 36



To prove that two people of the same nationality shook hands, we need to estimate the maximum number of handshakes between people of different nationalities.

For Phrase 1, we need to find the maximum number of handshakes between Ukrainians and Poles. We have 6 Ukrainians and 5 Poles, and each Ukrainian can shake hands with at most 5 Poles (since they cannot shake hands with themselves or with another Ukrainian), giving us a maximum of 6 x 5 = 30 handshakes.

However, each handshake is counted twice (once for each person involved), so we divide by 2 to get the maximum number of handshakes, which is (5 x 2) = 10.

For Phrase 2, we need to find the maximum number of handshakes between Ukrainians and Slovaks. We have 6 Ukrainians and 7 Slovaks, and each Ukrainian can shake hands with at most 7 Slovaks, giving us a maximum of 6 x 7 = 42 handshakes.

However, each handshake is counted twice, so we divide by 2 to get the maximum number of handshakes, which is (6 x 2) = 12.

For Phrase 3, we need to find the maximum number of handshakes between Poles and Slovaks. We have 5 Poles and 7 Slovaks, and each Pole can shake hands with at most 7 Slovaks, giving us a maximum of 5 x 7 = 35 handshakes.

However, each handshake is counted twice, so we divide by 2 to get the maximum number of handshakes, which is (7 x 2) = 12.

For Phrase 4, we need to find the total number of handshakes between people of different nationalities. We add up the maximum number of handshakes between Ukrainians and Poles, Ukrainians and Slovaks, and Poles and Slovaks, which gives us (10 + 12 + 12) = 34.

However, we need to remember that each handshake is counted twice, so we divide by 2 to get the total number of handshakes, which is (34/2) = 17.

Since we are given that the total number of handshakes is 110, which is greater than the total number of handshakes between people of different nationalities (17), we can conclude that there must be at least one pair of people who have the same nationality and shook hands. Therefore, we have proven that if 110 handshakes were exchanged in total, then two people of the same nationality shook hands.

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This probability is less than the significance level of 0.01, we again reject the null hypothesis.

We can conclude that the exact binomial method leads to the same conclusion as the normal approximation method.

To find the point estimate for p, we divide the number of individuals with PTSD in the sample by the total sample size:

[tex]\hat{p}[/tex] = 86/746

= 0.1154

The point estimate for p is 0.1154 or approximately 11.54%.

To construct a 95% confidence interval for p, we will use the following formula:

[tex]\hat{p}[/tex]  [tex]\pm z*\sqrt{(\hat{p} (1-\hat{p})/n)}[/tex]

Where z is the z-score for the desired confidence level (1.96 for 95% confidence), [tex]\hat{p}[/tex]  is the point estimate for p,

and n is the sample size.

Substituting the values given in the problem, we get:

0.1154 ± 1.96sqrt(0.1154(1-0.1154)/746)

The 95% confidence interval for p is (0.089, 0.142), meaning that we are 95% confident that the true proportion of individuals with PTSD in the population being treated in Veterans Affairs primary care clinics falls between 8.9% and 14.2%.

To construct a 99% confidence interval for p, we will use the same formula but with a z-score of 2.576 (from a standard normal distribution table).

0.1154 ± 2.576sqrt(0.1154(1-0.1154)/746)

The 99% confidence interval for p is (0.079, 0.152). This interval is wider than the 95% confidence interval because we are more confident that the true proportion falls within this interval.

The null hypothesis for this test is that the proportion of individuals with PTSD among those being treated in Veterans Affairs primary care clinics is equal to 7%, the prevalence reported in the prior study.

The alternative hypothesis is that the proportion is not equal to 7%.

Using the normal approximation to the binomial distribution, we can calculate the test statistic:

z = ([tex]\hat{p}[/tex] - 0.07) / [tex]\sqrt{(0.07 * 0.93 / 746)}[/tex]

Substituting the values, we get:

[tex]z = (0.1154 - 0.07) / \sqrt{(0.07 * 0.93 / 746) } = 3.05[/tex]

The p-value associated with this test statistic is approximately 0.0023. This means that if the true proportion of individuals with PTSD in the population being treated in Veterans Affairs primary care clinics is equal to 7%, we would expect to observe a sample proportion as extreme as 0.1154 in only 0.23% of all possible samples.

Since the p-value is less than the significance level of 0.01, we reject the null hypothesis.

This means that we have evidence to suggest that the proportion of individuals with PTSD among those being treated in Veterans Affairs primary care clinics is different from 7%.

To conduct the test using the exact binomial method, we can use software or a binomial distribution table to calculate the probability of getting 86 or more individuals with PTSD in a sample of 746 if the true proportion is 7%.

Using a binomial distribution table, we find that the probability of getting 86 or more individuals with PTSD out of 746 if the true proportion is 7% is approximately 0.

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The minimum length of ribbon that Bradley needs is 50.24 cm.

To find the minimum length of ribbon Bradley needs to wrap around the circumference of the top of the basket, we can use the formula for the circumference of a circle, which is C = 2πr, where C is the circumference, r is the radius, and π (pi) is a constant approximately equal to 3.14.

So, for Bradley's basket, the radius is 8 centimeters, and the circumference would be:

C = 2πr = 2π(8) = 16π = 50.24

Therefore, the minimum length of ribbon Bradley needs is 50.24 centimeters. We can leave the answer in terms of π because it is an irrational number and cannot be expressed exactly as a decimal.

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The probability of taking at least four trials to find the first defective light bulb is 0.073 (rounded to 3 decimal places).

To find the probability of taking at least four trials to find the first defective light bulb, we can use the geometric distribution formula:
P(X >= k) = (1-p)^(k-1) * p
Where X is the number of trials needed to find the first defective light bulb, p is the probability of finding a defective bulb on any given trial, and k is the minimum number of trials required.
In this case, we want to find the probability of taking at least four trials, so k = 4. We also know that the probability of finding a defective bulb on any given trial is 0.1 (since there is a 10% chance of any given bulb being defective). Therefore, we can plug in these values:
P(X >= 4) = (1-0.1)^(4-1) * 0.1
P(X >= 4) = 0.729 * 0.1
P(X >= 4) = 0.0729
So the probability of taking at least four trials to find the first defective light bulb is 0.073 (rounded to 3 decimal places).

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Answer:

Solving for y

exact form for y = 4/3

Demical form for y = 1/3

Mixed number for y = 1 1/3

( BTW all of these answers all right it juts depends on what exactly there asking for, for example if there asking for the exact form for y it would be 4/3 )

The range of the test scores in Mr. Miller's math class is 42.

Mathematically, the range refers to the difference between the highest value and the lowest value in a data set.

The range is computed by subtraction of the lowest value from the highest value.

Mr. Miller can use the range to measure the spread or dispersion of the test scores.

Test Scores:

52, 61, 69, 76, 82, 84, 85, 90, 94

Highest score = 94

Lowest score = 52

Range = 42 (94 - 52)

Thus, we can conclude that for the math students in Mr. Miller's class, the range of their test scores is 42.

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Jon will pay $3476.88 for the vacation package after the 25% discount and 9% resort charge are applied.

If the vacation package is obtainable for 25% off, then Jon will pay 75% of the original price. To discover the price after the discount, we can now multiply the original fee via 0.75:

Discounted charge = 0.75 x $4250 = $3187.50

Next, we need to add the 9% resort charge to the discounted fee. To do that, we are able to do multiply the discounted price by using 1.09:

Total cost = $3187.50 x 1.09 = $3476.88

Therefore, Jon will pay $3476.88 for the vacation package.

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The probability of NOT drawing a face card from a standard deck of 52 cards is 10 over 13.

First determine the total number of face cards and non-face cards in a standard deck of 52 cards.                                   In a standard deck, there are 12 face cards (3 face cards per suit: Jack, Queen, and King, and 4 suits: Hearts, Diamonds, Clubs, and Spades). This means there are 52 - 12 = 40 non-face cards.

Now, we'll calculate the probability of NOT drawing a face card:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability of NOT drawing a face card = (Number of non-face cards) / (Total number of cards)
Probability = 40 / 52

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (4):
Probability = (40/4) / (52/4)
Probability = 10 / 13

So, the probability of NOT drawing a face card from a standard deck of 52 cards is 10/13. Your answer: 10 over 13.

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The process of using sample statistics to draw conclusions about population parameters is called statistical inference. In statistical inference, we use the information obtained from a sample to make inferences about the characteristics of a larger population.

The sample statistics provide an estimate of the corresponding population parameters, and the goal is to make the most accurate inference possible.
The significance level, also known as alpha, is a pre-determined threshold that is used to determine the level of evidence required to reject the null hypothesis. This threshold is typically set at 0.05 or 0.01, depending on the level of certainty required.

In summary, statistical inference involves using sample statistics to make inferences about population parameters, and the significance level is a critical component of this process as it helps to determine the level of evidence required to reject the null hypothesis.
Statistical inference involves making choices about the sampling method and using collected data from a sample to make conclusions about a larger population. Sample statistics are calculations derived from a subset of the population, while population parameters are the true values for the entire population.

The significance level, typically denoted by α (alpha), is a predetermined threshold used to determine if a result is statistically significant. In hypothesis testing, if the calculated probability (p-value) is less than the significance level, we reject the null hypothesis and conclude that there is a statistically significant difference between the observed sample statistics and the expected population parameters.
In summary, statistical inference is the process of using sample statistics to draw conclusions about population parameters. It involves making choices regarding sampling methods and significance levels. The statement provided is False since statistical inference is not limited to finding the significance level.

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Percentage of change is increase and the percentage of change is 133.3%.

Change in percentage = changed percent × 100 / initial value

3/8 is changed into 7/8.

Change in value = 7/8 - 3/8 = 1/2

Change in percentage = 1/2 / 3/8 × 100

= 133.3%

Hence the percentage of change is increase and the percentage of change is 133.3%.

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The area of the triangle with the given vertices is approximately 13.95 square units.

To find the area of the triangle with the given vertices, we can use calculus to calculate the magnitude of the cross-product of two of its sides. Specifically, we can use the vectors formed by two pairs of vertices and take their cross-product to find the area.

Let's choose the vectors formed by the points (0,0) and (3,2) as well as (0,0) and (1,6). We'll call these vectors u and v, respectively:

u = <3, 2>

v = <1, 6>

To take the cross product of these vectors, we can use the formula:

|u x v| = |u| |v| sin(theta)

where |u| and |v| are the magnitudes of the vectors, and theta is the angle between them.

To find the angle between u and v, we can use the dot product formula:

u · v = |u| |v| cos(theta)

Solving for cos(theta), we get:

[tex]$\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\lvert\mathbf{u}\rvert \lvert\mathbf{v}\rvert} = \frac{(3 \cdot 1) + (2 \cdot 6)}{\sqrt{3^2 + 2^2} \sqrt{1^2 + 6^2}} = \frac{21}{\sqrt{13} \sqrt{37}}$[/tex]

We can then use the Pythagorean identity to find sin(theta):

[tex]$\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2}$[/tex]

Finally, we can plug in the values we've found to the formula for the magnitude of the cross-product:

[tex]$\lvert\mathbf{u} \times \mathbf{v}\rvert = \lvert\mathbf{u}\rvert \lvert\mathbf{v}\rvert \sin(\theta) = \sqrt{3^2 + 2^2} \sqrt{1^2 + 6^2} \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2}$[/tex]

Evaluating this expression gives us the area of the triangle:

[tex]$\lvert\mathbf{u} \times \mathbf{v}\rvert = 9 \sqrt{37} \sqrt{1 - \left(\frac{21}{\sqrt{13}\sqrt{37}}\right)^2} \approx 13.95$[/tex]

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The probability that your partner has the other three aces given that you have one ace is approximately 0.000037 or 0.0037%. This is a very low probability, so it is unlikely that your partner has the other three aces.

We can use Bayes' theorem to solve this problem. Let A be the event that your partner has the other three aces, and B be the event that you have one ace. Then we want to find P(A|B), the probability that your partner has the other three aces given that you have one ace.

Using Bayes' theorem, we have:

P(A|B) = P(B|A) * P(A) / P(B)

We know that P(B|A) is 1/3, since your partner must have all three aces if you have one ace. We also know that P(A) is the probability that your partner has all three aces in a randomly dealt hand of 39 cards (52 cards - 13 cards dealt to you), which is:

P(A) = (4/39) * (3/38) * (2/37) = 0.000038

To find P(B), the probability that you have one ace, we can use the law of total probability:

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

The probability that you have one ace given that your partner has all three aces is zero, so P(B|not A) is the probability of getting exactly one ace in a hand of 39 cards:

P(B|not A) = (4/39) * (35/38) * (34/37) * (33/36) * ... * (28/16) * (24/15) * (23/14) * (22/13) = 0.03833

where we multiply the probabilities of not getting an ace (35/38, 34/37, etc.) by the probability of getting an ace (4/36) for each card dealt after the first ace.

We also know that P(not A) is the complement of P(A), which is 1 - P(A).

Putting it all together, we have:

P(A|B) = (1/3) * (0.000038) / [ (1/3) * (0.000038) + (2/3) * (0.03833) ]

Simplifying, we get:

P(A|B) = 0.000037

So the probability that your partner has the other three aces given that you have one ace is approximately 0.000037 or 0.0037%. This is a very low probability, so it is unlikely that your partner has the other three aces.

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The graph is  both graph A and graph B, because both curves are above the horizontal axis, and their areas are positive

Density curves are visuals that demonstrate the probability distribution of a data set.

It is a liquid, uninterrupted line that illustrates the variability of a constant haphazard element - with the entire locality below the curve accounting for 1.

In other words, the region underneath the curve denotes the likelihood of registering a precise or scope of values inside the depth of the grouping.

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The value of the probability of choosing a green lollipop and a purple lollipop is, 8 / 45

We have to given that;

One bag contains 3 red lollipops and 2 green lollipops.

And, The other bag contains four purple lollipops and five blue lollipops.

Hence, The probability of choosing a green lollipop is,

P₁ = 2 / 5

And, The probability of choosing a purple lollipop is,

P₂ = 4 / 9

Thus, The value of the probability of choosing a green lollipop and a purple lollipop is,

P = P₁ ×  P₂

P = 2/5 × 4/9

P = 8/45

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The number 0.909009000900009000009... is irrational. To determine if a number is rational or irrational, we need to see if it can be expressed as a ratio of two integers. However, this number does not repeat in a regular pattern, so we cannot express it as a fraction. Therefore, it is irrational.

In general, if a decimal number does not repeat in a regular pattern, it is likely to be irrational.
The given number, 0.909009000900009000009..., is an irrational number.

The reason for this answer is that a rational number can be expressed as a fraction, where both the numerator and the denominator are integers and the denominator is not zero. However, this number has a non-terminating, non-repeating decimal pattern, which makes it impossible to represent it as a fraction. Thus, it is irrational.

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The area of triangle RST is 27 square units.

Option A is the correct answer.

We have,

To find the area of the triangle RST, we can use the formula:

Area = 1/2 x base x height

where the base is the distance between any two of the vertices, and the height is the perpendicular distance from the third vertex to the line containing the base.

Let's take RS as the base.

The distance between R and S is 2 + 7 = 9 units.

To find the height, we need to determine the equation of the line containing the base RS, and then find the distance from vertex T to this line.

The slope of the line RS is:

(y2 - y1)/(x2 - x1) = (-7 - 2) / (-9-(-9)) = -9/0,

which is undefined.

This means that the line is vertical and has the equation x = -9.

The perpendicular distance from T to the line x = -9 is simply the horizontal distance between T and the point (-9,-7), which is 6 units.

Therefore,

The area of triangle RST is:

Area = 1/2 x base x height = 1/2 x 9 x 6 = 27 square units.

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Answer:

  (c)  x = 0.39 and x = -3.89

Step-by-step explanation:

You want the solutions to the quadratic equation ...

  2x² +7x = 3.

The roots of the equation ...

  x² +bx +c = 0

have a sum of -b and a product of c.

Subtracting 3 and dividing the equation by 2, we have ...

  2x² +7x -3 = 0 . . . . . . . . subtract 3

  x² +3.5x -1.5 = 0 . . . . . . divide by 2

This tells us the sum of the roots is -3.5.

Answer choice C has that sum: x = 0.39, x = -3.89.

__

Additional comment

The sums of the answer choices are ...

  0.60 -2.60 = -2.00

  -0.60 +2.60 = 2.00

  0.39 -3.89 = -3.50

  -0.39 +3.89 = 3.50

Sometimes, checking the offered choices is the simplest way to find the answer.

Here, checking the sum gives the best discriminator of right from wrong. The products are all near -1.5, so that is less helpful.

We can see the relation by considering the factored form:

  (x -p)(x -q) = x² -(p+q)x +pq . . . . . . where p and q are the roots

<95141404393>

The function graphed on the axes above should be expressed as a piecewise function as follows;

f(x) = -3x - 8    {x ≤ -2}

    = 6x - 17     {x > 3}

In order to determine the piecewise function, we would determine an equation that represent each of line shown on the graph. Therefore, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (4 + 2)/(-4 + 2)

Slope (m) = 6/-2

Slope (m) = -3.

At data point (-2, -2) and a slope of -3, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y + 2 = -3(x + 2)  

y = -3x - 8

For the second line, we have:

Slope (m) = (7 - 1)/(4 - 3)

Slope (m) = 6/1

Slope (m) = 6.

At data point (3, 1) and a slope of 6, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 1 = 6(x - 3)  

y = 6x - 17

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Answer:

[tex] {7}^{6} [/tex]

The base is 7, and the exponent is 6.

The graph is misleading because the y values do not start from the origin

The graph represents the given parameter where

Examining the lengths of the bars with the values, we can see that

The y values do not start from the origin

This is because difference between the bars do not show the correct representation

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To calculate the selling price of the house, we can use the formula for a mortgage:

M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1]

Where:
M = monthly payment
P = principal (selling price)
i = interest rate per month (4.5%/12)
n = total number of payments (30 years x 12 months)

We know that the monthly payment is $811 and the total number of payments is 30 years x 12 months = 360 months. So we can solve for the principal:

$811 = P [ (0.045/12) (1 + 0.045/12)^360 ] / [ (1 + 0.045/12)^360 – 1]

$811 = P [ 0.00375 (1 + 0.00375)^360 ] / [ (1 + 0.00375)^360 – 1]

$811 = P [ 0.00375 (3.8113) ] / [ 3.8113 – 1]

$811 = P [ 0.014287 ]

P = $56,732.77

Therefore, the selling price of the house was $56,732.77.

To calculate the total interest paid over 30 years, we can use the formula:

Total interest = (monthly payment x total number of payments) - principal

Total interest = ($811 x 360) - $13,694

Total interest = $292,740 - $13,694

Total interest = $279,046

Therefore, they will pay a total of $279,046 in interest over 30 years.

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You would expect have 5 CDs to be rap in the next 30 CDs

From the question, we have the following parameters that can be used in our computation:

pop 65

rock 10

rap 15

This means that we have the following proportion

Rap = 15/(65 + 10 + 15)

Evaluate

Rap = 15/90

So, we have

Rap = 1/6

Considering loaning 30 CDs out, we have

Rap = 1/6 * 30

Rap = 5

Hence, the expected values of rap CDs is 5

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The length of FH measures as 18 unit.

A quadrilateral in which opposite sides are parallel is called a parallelogram, a parallelogram is always a quadrilateral but a quadrilateral can or cannot be a parallelogram.

We are given the diagonals as;

FH = 4z -9 + 2z

EG = 3w +w + 8

Therefore, we know that the diagonal of the parallelogram bisect each other.

FJ = JH

4z -9  = 2z

4z - 2z = 9

2z = 9

z = 9/2

Then FH = 4z -9 + 2z

FH = 4(9/2) -9 + 2(9/2)

FH = 18 - 9 + 9

FH = 18

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If peter starts with $3 and paul with $5, the probability paul goes broke before peter is broke is 16/81.

Let's first consider the probability that Peter goes broke before Paul. For Peter to go broke, he needs to lose all of his $3 in the first two games. The probability of this happening is:

(2/3)² = 4/9

If Peter goes broke, then Paul has won $2 and has $7 left. Now, the game is between Paul's $7 and Peter's $1. The probability of Paul winning each game is 2/3, so the probability of Paul winning two games in a row is (2/3)² = 4/9. Therefore, the probability of Paul winning two games in a row and going broke before Peter is broke is:

4/9 x 4/9 = 16/81

So the probability that Paul goes broke before Peter is broke is 16/81.

The probability that Peter goes broke before Paul is 4/7.

For similar question on geometric distribution:

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If pmf of a random variable is given by 4 f(X=n)= n(n+1)(n+2)

the answer to part (a) is:

Ë F(X = n) = 9n(n+1)

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

a. To show that the cumulative distribution function (CDF) F(X) satisfies Ë F(X = n)=1, we need to show that the sum of the probabilities of all possible values of X is equal to 1.

The probability mass function (PMF) is given by:

f(X=n) = 4n(n+1)(n+2)

The CDF is defined as:

F(X=n) = P(X ≤ n)

We can calculate F(X=n) by summing up the probabilities of all values less than or equal to n:

F(X=n) = Σ f(X=i), for i = 0 to n

Substituting the given PMF:

F(X=n) = Σ 4i(i+1)(i+2), for i = 0 to n

Expanding the sum:

F(X=n) = 4(0)(1)(2) + 4(1)(2)(3) + 4(2)(3)(4) + ... + 4n(n+1)(n+2)

F(X=n) = 4 [ (0)(1)(2) + (1)(2)(3) + (2)(3)(4) + ... + (n)(n+1)(n+2) ]

Notice that the sum inside the brackets is a telescoping sum, which can be simplified as:

[(k-1)k(k+1) - (k-2)(k-1)k] = 3k(k-1)

Thus,

F(X=n) = 4 [ 3(0)(-1) + 3(1)(0) + 3(2)(1) + ... + 3(n)(n-1) ]

F(X=n) = 4 [ 3(0 + 1 + 2 + ... + (n-1)) ]

F(X=n) = 4 [ 3(n-1)n/2 ]

F(X=n) = 6n² - 6n

Therefore, Ë F(X = n) is given by:

Ë F(X = n) = Σ F(X=n) * P(X=n), for all n

Substituting the given PMF:

Ë F(X = n) = Σ [ 6n² - 6n ] * 4n(n+1)(n+2), for all n

Expanding the sum and simplifying:

Ë F(X = n) = 24 [ (n+2)(n+1)n(n-1)/4 - (n+1)n(n-1)(n-2)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2)/4 - (n-2)(n-1)n(n+1)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1) ] / 4

Ë F(X = n) = 6n(n+1)(n+2) - 6n(n+1)(n-1) / 4

Ë F(X = n) = 6n(n+1)[ (n+2) - (n-1) ] / 4

Ë F(X = n) = 6n(n+1) * 3 / 4

Ë F(X = n) = 9n(n+1)/2

Substituting n = 0 and n = ∞ to get the bounds of the sum, we get:

E[X] = 2(0)(5(0)+8) / 3 + 2(∞)(∞+1)(5(∞)+8) / 3

Since the second term diverges to infinity, we can conclude that the expected value of X does not exist (i.e., it is undefined).

Therefore, the answer to part (a) is:

Ë F(X = n) = 9n(n+1)/

To learn more about probability from the given link:

https://brainly.com/question/30034780

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If pmf of a random variable is given by 4 f(X=n)= n(n+1)(n+2)

the answer to part (a) is:

Ë F(X = n) = 9n(n+1)

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

a. To show that the cumulative distribution function (CDF) F(X) satisfies Ë F(X = n)=1, we need to show that the sum of the probabilities of all possible values of X is equal to 1.

The probability mass function (PMF) is given by:

f(X=n) = 4n(n+1)(n+2)

The CDF is defined as:

F(X=n) = P(X ≤ n)

We can calculate F(X=n) by summing up the probabilities of all values less than or equal to n:

F(X=n) = Σ f(X=i), for i = 0 to n

Substituting the given PMF:

F(X=n) = Σ 4i(i+1)(i+2), for i = 0 to n

Expanding the sum:

F(X=n) = 4(0)(1)(2) + 4(1)(2)(3) + 4(2)(3)(4) + ... + 4n(n+1)(n+2)

F(X=n) = 4 [ (0)(1)(2) + (1)(2)(3) + (2)(3)(4) + ... + (n)(n+1)(n+2) ]

Notice that the sum inside the brackets is a telescoping sum, which can be simplified as:

[(k-1)k(k+1) - (k-2)(k-1)k] = 3k(k-1)

Thus,

F(X=n) = 4 [ 3(0)(-1) + 3(1)(0) + 3(2)(1) + ... + 3(n)(n-1) ]

F(X=n) = 4 [ 3(0 + 1 + 2 + ... + (n-1)) ]

F(X=n) = 4 [ 3(n-1)n/2 ]

F(X=n) = 6n² - 6n

Therefore, Ë F(X = n) is given by:

Ë F(X = n) = Σ F(X=n) * P(X=n), for all n

Substituting the given PMF:

Ë F(X = n) = Σ [ 6n² - 6n ] * 4n(n+1)(n+2), for all n

Expanding the sum and simplifying:

Ë F(X = n) = 24 [ (n+2)(n+1)n(n-1)/4 - (n+1)n(n-1)(n-2)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2)/4 - (n-2)(n-1)n(n+1)/4 ]

Ë F(X = n) = 24 [ (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1) ] / 4

Ë F(X = n) = 6n(n+1)(n+2) - 6n(n+1)(n-1) / 4

Ë F(X = n) = 6n(n+1)[ (n+2) - (n-1) ] / 4

Ë F(X = n) = 6n(n+1) * 3 / 4

Ë F(X = n) = 9n(n+1)/2

Substituting n = 0 and n = ∞ to get the bounds of the sum, we get:

E[X] = 2(0)(5(0)+8) / 3 + 2(∞)(∞+1)(5(∞)+8) / 3

Since the second term diverges to infinity, we can conclude that the expected value of X does not exist (i.e., it is undefined).

Therefore, the answer to part (a) is:

Ë F(X = n) = 9n(n+1)/

To learn more about probability from the given link:

brainly.com/question/30034780

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