Charles has two six-sided die. one of the die is fair, and the other die is biased so that it comes up six with probability 2 3 and each of the other five sides has probability 15 1 . charles chooses one of the two dice at random and rolls it three times. given that the first two rolls are both sixes, the probability that the third roll will also be a six is p q, where p and q are relatively prime positive integers. find p q. (2014 aime ii problem 6 )

Thus, the number of 6-digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is 20.

What is the number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated?There are 1200 6-digit numbers that can be formed using the digits 0, 1, 2, 5, 7, and 9.Using the formula for a 2-digit number, namely A and B, the number must satisfy A - B = 0,11 or -11.

There is a pattern, therefore, there are 100 numbers that satisfy each. It's because the numbers 0, 1, 2, 5, 7, and 9 are such that the sum of the odd digits minus the sum of the even digits is always either 0, 11, or -11. This is only possible if there is an even number of odd digits or an even number of even digits.

There are 6C3 ways to pick 3 odd digits from {1,5,7,9} and 3C3 ways to pick 3 even digits from {0,2}. By taking the product of these values, we can find the total number of possibilities:[tex]6C3 × 3C3 = 20 × 1 = 20[/tex]

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the 90% confidence interval for the proportion of pain reduction for the vitamin A oil group is (0.275, 0.465) , placebo group is (0.119, 0.301).

Part A: An appropriate design for the new study would be a randomized controlled trial. The treatments used in this study would be vitamin A oil and a placebo. The method of treatment assignment would be randomized, with patients being randomly assigned to receive either vitamin A oil or the placebo. The variables that should be measured include the level of pain reduction after treatment, the age of the patients, the severity of the pinched nerve, and any side effects experienced by the patients.

Part B: If the study consisted of 100 young patients and 100 old patients instead of 200 patients aged 20–40, the study design would need to be modified. Age is an important variable that could affect the results of the study. To control for this variable, the study could be designed to have two separate treatment groups: one group of young patients and one group of old patients. This would allow for the comparison of pain reduction between the two age groups and would provide more accurate results.

Part C: To construct 90% confidence intervals for the proportion of pain reduction for each treatment group, we can use the following formula:

CI = p ± z*(√(p(1-p)/n))

where p is the proportion of patients reporting pain reduction, z is the critical value from the standard normal distribution for a 90% confidence interval (which is 1.645), and n is the total number of patients in the treatment group.

For the vitamin A oil group:

p = 0.37

n = 100

z = 1.645

CI = 0.37 ± 1.645*(√(0.37(1-0.37)/100))

CI = 0.37 ± 0.095

CI = (0.275, 0.465)

Therefore, the 90% confidence interval for the proportion of pain reduction for the vitamin A oil group is (0.275, 0.465).

For the placebo group:

p = 0.21

n = 100

z = 1.645

CI = 0.21 ± 1.645*(√(0.21(1-0.21)/100))

CI = 0.21 ± 0.091

CI = (0.119, 0.301)

Therefore, the 90% confidence interval for the proportion of pain reduction for the placebo group is (0.119, 0.301).

Part D: Based on the data collected, we can conclude that vitamin A oil is effective in reducing pain levels in patients suffering from pinched nerves in the cervical region of the back. The 90% confidence interval for the proportion of pain reduction for the vitamin A oil group (0.275, 0.465) does not overlap with the 90% confidence interval for the proportion of pain reduction for the placebo group (0.119, 0.301), indicating a statistically significant difference between the two groups. However, further research may be needed to confirm these results and to investigate any potential side effects of vitamin A oil treatment.

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The degrees of freedom associated with a pooled-variance two-sample t-test or confidence interval in the given case would be 48.

When a pooled-variance two-sample t-test or confidence interval is to be conducted on two groups, the degrees of freedom is calculated using the following formula: df = n1 + n2 - 2, where n1 and n2 are the sample sizes of the two groups being compared.

In the given scenario, data is collected from 20 random luxury hotels in New York and 30 random luxury hotels in Los Angeles. Therefore, the degrees of freedom for a pooled-variance two-sample t-test or confidence interval would be:df = 20 + 30 - 2df = 48. Hence, the degrees of freedom associated with a pooled-variance two-sample t-test or confidence interval are 48.

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In statistics, the mean is a measure of central tendency that represents the average value of a set of data. It is calculated by adding up all the values in the data set and dividing by the total number of values. The mean is also known as the arithmetic mean or average.

The formula for the mean is:

mean = (sum of all values) / (total number of values)

1.To find the mean of these scores, we add them up and divide by the total number of scores:

Mean = (83 + 97 + 85 + 84 + 96 + 80 + 80 + 87 + 91) / 9 = 86

2.To find the median, we first need to put the scores in order from lowest to highest:

80, 80, 83, 84, 85, 87, 91, 96, 97

The median is the middle score, which in this case is 85.

3.To find the mode, we look for the score that occur most frequently. In this case, there are two modes: 80 and 85, each occurring twice.

4.To find the range, we subtract the lowest score from the highest score:

Range = 97 - 80 = 17

Therefore, the mean is 86, the median is 85, the mode(s) are 80 and 85, and the range is 17.

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Answer:

1/24

Step-by-step explanation:

1 ft = 12 in

8 X 12 = 96

96/4 = 24

1/24 is the scale factor

Answer: 24:1 (The tiger : The tiger model)

Step-by-step explanation:

1 feet =12 inch

so 8 feet=96 inch

The tiger : The tiger model= 96:4= 24:1

The probability that the grocer arranges the fruits in alternating order is 5/16.

It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

There are only two ways that the fruits can be arranged in alternating order: apple-orange-apple-orange-apple-orange-apple-orange or orange-apple-orange-apple-orange-apple-orange-apple.

Let's consider the first case where the first fruit is an apple. The probability of this happening is 1/2 since there are 10 total fruits and 5 of them are apples. Once the first fruit is chosen, there are 4 apples and 5 oranges left. The next fruit must be an orange, which has probability 5/9 of occurring since there are now 9 fruits left and 5 of them are oranges. The third fruit must be an apple, which has probability 4/8 (or 1/2) of occurring since there are 8 fruits left and 4 of them are apples. Continuing in this way, we get the following probability:

(1/2) * (5/9) * (4/8) * (4/7) * (3/6) * (3/5) * (2/4) * (2/3) * (1/2) * 1

The second case where the first fruit is an orange is the same as the first case but with the roles of apples and oranges reversed. So the probability of arranging the fruits in alternating order is:

(1/2) * (4/9) * (5/8) * (3/7) * (4/6) * (2/5) * (3/4) * (1/3) * (2/2) * 1

Simplifying both expressions, we get:

(5!/(4!1!)) * (4!/(2!2!)) * 1/(2^5) + (5!/(4!1!)) * (4!/(2!2!)) * 1/(2^5)

= 10/32 + 10/32

= 5/16

Therefore, the probability that the grocer arranges the fruits in alternating order is 5/16.

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Answer:

the value of x when f(x) = 15 is 3.5.

Step-by-step explanation:

We are given that:

f(x) = 1 + 4x

We need to find the value of x when f(x) = 15.

So we can set up the equation:

1 + 4x = 15

Subtracting 1 from both sides, we get:

4x = 14

Dividing both sides by 4, we get:

x = 14/4

Simplifying, we get:

x = 3.5

Therefore, the value of x when f(x) = 15 is 3.5.

Answer:

[tex] < 2 \: and \: <7[/tex]

8 people in Malik's apartment building like to eat macaroni and cheese or cookies or both

Of the people in Malik's apartment building, 4 like to eat macaroni and cheese, and 7 like to eat cookies.

Three people like to eat both macaroni and cheese and cookies.

To determine how many people like to eat macaroni and cheese or cookies or both, we will employ the principle of inclusion-exclusion.

In Malik's apartment building, to find out how many people like to eat macaroni and cheese, or cookies, or both, we can follow these steps:
Identify the number of people who like macaroni and cheese (4 people) and cookies (7 people).
Determine the number of people who like both macaroni and cheese and cookies (3 people).
Calculate the total number of people who like either macaroni and cheese or cookies by adding the number of people who like macaroni and cheese (4 people) and the number of people who like cookies (7 people).
4 people + 7 people = 11 people
Subtract the number of people who like both macaroni and cheese and cookies (3 people) from the total calculated in Step 3 to avoid double-counting those who like both.
11 people - 3 people = 8 people

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The sum of the squares of JE and EN is approximately 32. Therefore, we can conclude that the triangle formed by J, E, and N is indeed an isosceles right triangle.

An isosceles right triangle is a triangle that has two sides of equal length and one right angle. In this case, we are given the coordinates of three points J(-4,1), E(-2,-3), and N(2,-1) and we need to determine if the triangle formed by these points is an isosceles right triangle.

Firstly, we need to calculate the length of each side of the triangle. We can use the distance formula, which states that the distance between two points (x1, y1) and (x2, y2) is given by the square root of ((x2-x1)^2 + (y2-y1)^2).

Using this formula, we can find that the length of JE is approximately 4.472, the length of EN is approximately 4.472, and the length of JN is approximately 5.657.

Since two sides of the triangle (JE and EN) have the same length, the triangle is indeed isosceles. To determine if it is also a right triangle, we need to check if one of the angles is a right angle.

We can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we can see that JN is the longest side and its length squared is approximately 32. Therefore, we need to check if the sum of the squares of JE and EN is also approximately 32.

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The radius of each circle is approximately 3.46 units.

Since the two circles are congruent and tangent to each other, their centers are located on the same line perpendicular to the side of the square.

Let the center of each circle be O and the point where the circles are tangent be T. Then, the radius of each circle is equal to the distance from O to T.

Drawing a line from O to T, we can see that it forms a right triangle with one leg equal to the radius of the circle and the other leg equal to half the side length of the square. Using the Pythagorean theorem, we can solve for the radius:

[tex]r^2 + (2)^2 = (4)^2r^2 = 12\\[/tex]

r ≈ 3.46

Therefore, the radius of each circle is approximately 3.46 units, rounded to the nearest hundredth.

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Answer:

x = 10

Step-by-step explanation:

To solve for x, we need to get x by itself.

Whatever we do to one side, we do it to the other side to keep it balanced.

3x + 15 = 45

     - 15 = -15       (Subtract 15 from both sides)

3x        =   30               (Divide both sides by 3)

3          =    3

x = 10

Answer:

Step-by-step explanation:

The surface area of the triangular prism is 168 square feet.

Surface area is the total area that the surface of a three-dimensional object occupies in space. It is a measure of the amount of material that would be needed to cover the entire surface of the object.

To find the surface area of the triangular prism, we need to find the area of each of its faces and add them together.

The triangular faces have an area of [tex]\frac{1}{2}bh[/tex], where b is the base and h is the height. In this case, the triangular faces have a base of 6 feet and a height of 4 feet, so their area is:

=>[tex]\frac{1}{2}\times6\times4=12ft^2[/tex]

There are two triangular faces, so their total area is [tex]2\times12=24ft^2[/tex]

The rectangular faces have an area of lw, where l is the length and w is the width. In this case, the rectangular faces have dimensions of 9 feet by 5 feet, so their area is:

=> [tex]9\times5=45ft^2[/tex]

There are two rectangular faces, so their total area is

=> [tex]2\times45=90ft^2[/tex]

Finally, the base of the triangular prism is a rectangle with dimensions of 6 feet by 9 feet, so its area is:

=> [tex]6\times9 = 54 ft^2[/tex]

Adding up the areas of all the faces, we get:

=>[tex]24 ft^2 + 90 ft^2 + 54 ft^2 = 168 ft^2[/tex]

Therefore, the surface area of the triangular prism is 168 square feet.

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The ratio of the first term to the common difference is 1 is to 1 (1:1).

Let's say the first term of the arithmetic progression is 'a' and the common difference is 'd'.

Then the sum of the first five terms will be,

S5 = a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d) = 5a + 10d

And the sum of the first ten terms will be,

S10 = a + (a + d) + (a + 2d)+ ... + (a + 9d)

=10a + 45d

We have been given that sum of the first 10 terms is equal to four times the sum of the first 5 terms, i.e

S10 = 4S5

Substituting the expressions for S5 and S10 in terms of a and d, we get,

10a + 45d = 4(5a + 10d)

Simplifying this equation, we get,

10a + 45d = 20a + 40d

10d = 10a

d = a

So the ratio of the first term to the common difference is 1:1 or 1/1.

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A histogram is a basic statistical tool that graphically shows the frequency or number of observations of a particular value or within a specified group.

A histogram is a chart that displays data by dividing it into intervals, or "bins," and then counting how many data points fall into each bin. The x-axis of a histogram represents the range of values being analyzed, while the y-axis represents the frequency or count of values that fall within each bin.

Histograms are commonly used to illustrate the distribution of data. They provide a visual representation of the central tendency, variability, and shape of the data. In particular, histograms can reveal patterns in the data such as skewness, outliers, or gaps.

Histograms are frequently used in data analysis across many fields, including finance, biology, psychology, and engineering. They can be created using software programs such as Microsoft Excel, R, Python, or MATLAB.

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The given question is incomplete, the complete question is:

A(n)______ is a basic statistical tool that graphically shows the frequency or number of observations of a particular value or within a specified group

The area of the given figure is 44 units.

The side TU of the right triangle is 156.15 metres and the area of the triangle is  6168.07 metres squared.

The side GH of the right triangle is 182.33 metres and the area of the triangle is  8934.27 metres².

A right triangle is a triangle that has one of its angles as 90 degrees. The sum of angles in a triangle is 180 degrees.

The missing side of the right triangle can be found using Pythagoras's theorem.

Therefore,

a² + b² = c²

where

The area of the right triangle can be found using the formula as follows:

area of a triangle = 1 / 2 bh

where

Hence,

1.

TU² = 175² - 79²

TU² = 30625 - 6241

TU² = 24384

TU = √24384

TU = 156.153770368

TU = 156.15 metres

Area = 1 / 2 × 156.15 × 79

area = 6168.07392952

area = 6168.07 metres squared

2.

GH² = 207² - 98²

GH² = 42849 - 9604

GH = √33245

GH = 182.33211456

GH = 182.33 metres

Area of the triangle  = 1 / 2 × 98 × 182.33

Area of the triangle  = 8934.27361345

Area of the triangle  = 8934.27 metres²

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Answer:

x = 122

Step-by-step explanation:

the measure of the chord- chord angle 125° is half the sum of the arcs intercepted by the angle and its vertical angle, that is

125 = [tex]\frac{1}{2}[/tex] (x + 6 + x) ← multiply both sides by 2 to clear the fraction

250 = 2x + 6 ( subtract 6 from both sides )

244 = 2x ( divide both sides by 2 )

122 = x

Answer:

Step-by-step explanation:

-16-5x=-16. please help

-16 - 5x = -16

5x = -16 + 16

5x = 0

x = 0 : 5

x = 0

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

check

-16 - 5  × 0 = -16           (remember PEMDAS)

-16 = -16

the answer is good

Answer:

x = 0

Step-by-step explanation:

- 16- 5x = - 16

+ 16        + 16

   - 5x = 0

x = 0

Answer:

I'm sorry, it is unclear what you are asking. Can you please provide more context or clarify your question?

Step-by-step explanation:

Step-by-step explanation: I hope this helps.

Answer:

 [tex]3(x+1)(x+7)-(2x+5)^2[/tex] is never positive for any value of x, since it is equal to  [tex]-x^2+4x-4.[/tex]

To prove that  [tex]3(x+1)(x+7)-(2x+5)^2[/tex] is never positive, we can use algebraic manipulation and some basic concepts of quadratic equations.

First, let's expand the expression and simplify:

[tex]3(x+1)(x+7)-(2x+5)^2 = 3(x^2+8x+7)-(4x^2+20x+25)[/tex]

[tex]= 3x^2+24x+21-4x^2-20x-25[/tex]

[tex]= -x^2+4x-4[/tex]

Now we want to show that this expression is never positive for any value of x.

We can start by considering the discriminant of the quadratic expression -x²+4x-4, which is b²-4ac, where a = -1, b = 4, and c = -4. The discriminant is:

[tex]b^2-4ac = 4^2-4(-1)(-4) = 16-16 = 0[/tex]

Since the discriminant is zero, the quadratic equation [tex]-x^2+4x-4[/tex]  has a double root, which means that it touches the x-axis at exactly one point.

Since the coefficient of x² is negative, this means that the parabola opens downward, which implies that the function has a maximum value at the vertex. The x-coordinate of the vertex is given by -b/2a, which in this case is x [tex]= -4/-2 = 2[/tex] .

So, we can conclude that the expression   [tex]-x^2+4x-4[/tex]  is never positive for any value of x, since it has a maximum value of -4 at  [tex]x = 2[/tex] .

Therefore,  [tex]3(x+1)(x+7)-(2x+5)^2[/tex] is never positive for any value of x, since it is equal to [tex]-x^2+4x-4[/tex] .

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Hence [tex]3(x+1)(x+7)-(2x +5)^{2}[/tex] = -1 ([tex]x^{2} + 4x -4[/tex]) as the leading coefficient is -1, therefore is never positive.

Expanding the first and second term and then simplifying it we get,

To solve this equation first we need to simplify the equation and then we need to subtract the simplified term.

[tex](x+1)(x+7) = x^{2} +8x+7\\3(x^{2} +8x+7)=3x^{2} +24x+21\\(2x+5)^{2} =(2x+5)(2x+5)= 4x^{2} +20x+25\\(3x^{2} +24x+21) - (4x^{2} +20x+25)= -x^{2} + 4x - 4 =-1(x^{2} -4x+4)= -1((x+2)(x+2))[/tex]

Because it is always multiplied with -1

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If buses arrive at stop at "15-minute" intervals, then the probability that he waits more than 10 minutes for a bus is "1/3".

Let us denote the arrival-time of the passenger by X, where X is uniformly distributed between 7am and 7:30am. hence, the probability-density-function (pdf) of "X" is written as :

f(x) = 1/30, for 7am ≤ x ≤ 7:30am

f(x) = 0, otherwise

We observe that, the passenger will wait for more than 10 minutes for a bus only if he arrives between 7:00 a.m. and 7:05 a.m. , or between 7:15 a.m. and 7:20 a.m.

So, the probability for waiting time can be written as ;

⇒ [tex]\int\limits^5_0 {\frac{1}{30} } \, dx[/tex] + [tex]\int\limits^{20}_{15} {\frac{1}{30} } \, dx[/tex],

⇒ (1/30)(5 - 0) + (1/30)(20 - 15);

⇒ 1/3.

Therefore, the required probability is 1/3.

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The given question is incomplete, the complete question is

Buses arrive at a specified stop at 15-minute intervals starting at 7am. If a passenger arrives at the stop at a time that is uniformly distributed between 7am and 7:30am, find the probability that he waits more than 10 minutes.

Step-by-step explanation:

Area of the TWO ends are each 1/2 b*h = 1/2 * 16 * 6 =48 in^2

    TWO of them would be   2 * 48 = 96 in^2

Then there are 3 sides

   two are  10 * 12  = 120

       two would be    120 x 2 = 240 in^2

     and ONE side 16 * 12 = 192 in^2

Total area is then   96 + 240 + 192 = 528 in^2

Answer: Number 10 is you are a rectangle

Step-by-step explanation:

The measure of central tendency is the most appropriate for this data is median.

There are a few different measures of central tendency to choose from, but the most common ones are the mean, median, and mode.

The mean is the arithmetic average of the data. To find the mean, you would add up all the values and divide by the total number of values. However, the mean can be sensitive to outliers or extreme values, which could skew the results. In this case, there are no extreme values, but the data is not evenly distributed across the three groups, so the mean may not be the best measure of central tendency to use.

The median is the middle value of the data when it is arranged in order from smallest to largest.

This can be a better measure of central tendency when the data is not evenly distributed. In this case, we can find the median by arranging the data in order: 25, 125, 250. The middle value is 125, which means that half of the pet owners surveyed own fewer than 125 pets, and half own more than 125 pets.

So, the median might be a good choice of measure of central tendency in this case.

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Answer:

827.0 cm³

Step-by-step explanation:

The question we are told is to find the volume of water that can fill the paper cup. This means that we have to find the volume of a cylinder (the cup).

1) understand the equation for volume of a cylinder

The formula for the volume of a cylinder is πr²h

The radius would be half the diameter, which is 4.5, and h would be the height which is 13.

2) Solve

This means our working out would be...

3) Round

We have to round this to the nearest tenth which would be 827.0 or 827!

Hope this helps, have a lovely day! :)

Answer:

2.2 Cm

Step-by-step explanation:

Answer:

the answer is 1.9

Step-by-step explanation:

10 minus 8.1 equals 1.9 because the smallest crocus is 8.1 and the largest is 10 so you subtract those two to find the difference, thank me later!