Let X be the number of fixed points of a random permutation of n elements. A fixed point is an element that remains in the same position after the permutation. Thus, the probability that an element is fixed is 1/n, and the probability that it is not fixed is (n-1)/n.
Using the linearity of the expected value, we can calculate the expected value of X as:
E(X) = E(X1 + X2 + ... + Xn) = E(X1) + E(X2) + ... + E(Xn)
where Xi is the indicator random variable that is equal to 1 if the i-th element is fixed and 0 otherwise. Since the probability of an element being fixed is 1/n, we have E(Xi) = 1/n. Therefore,
E(X) = n * (1/n) = 1
To find the variance of X, we need to compute E(X^2) - E(X)^2. We can use the fact that X^2 = X1 + X2 + ... + Xn, where Xi is the indicator random variable that is equal to 1 if the i-th and j-th elements are both fixed and 0 otherwise. Then,
E(X^2) = E(X1 + X2 + ... + Xn)^2 = E(X1^2 + X2^2 + ... + Xn^2) + 2 E(X1X2 + X1X3 + ... + X(n-1)n)
Since there is only one way to fix two elements out of n, we have E(XiXj) = 1/(n(n-1)). Therefore,
E(X^2) = n * (1/n) + n(n-1) * (1/(n(n-1))) = 1 + 1/n
Finally, the variance of X is
Var(X) = E(X^2) - E(X)^2 = 1 + 1/n - 1^2 = 1/n
Therefore, the variance of the number of fixed elements of a randomly selected permutation of n elements is 1/n.
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How to seal a sofa is being sold for 26% of the regular price the sale price is $244.40 what is the regular price
Answer:
Step-by-step explanation:
You would divide 244.40 by 26%
You would get 63.544.... Now times the 63.544 to the remainder of the 26 from 100...which is 74. 63.544 times 74 equals 4702.256.
Now add the 244.40 to that and u get the answer of 4946.656.
Answer:
Good Luck
Step-by-step explanation:
Let the regular price be "x".
According to the problem, the sofa is being sold for 26% of the regular price, which means the sale price is:
Sale price = 0.26x
The problem also states that the sale price is $244.40, so we can write:
0.26x = 244.40
To solve for x, we need to isolate it on one side of the equation. We can do this by dividing both sides by 0.26:
x = 244.40 ÷ 0.26
x = 940
Therefore, the regular price of the sofa is $940.
What expresiokn is equialent to 42x-48
The expression that is equivalent to 42x - 48 is: 6(7x - 8)
How to factorize algebraic expressions?An algebraic expression is defined as a mathematical phrase that includes both variables, constants, coefficients, and algebraic operations.
An algebraic expression is also considered as a variable expression described using its terms, and operations on the terms. For example, x + 3 can be described as "3 more than x".
We want to factorize the expression 42x - 48
The number that is common to both numbers is 6 and as such we have the expression factorized as:
6(7x - 8)
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a triangle is conventionally used in a process flowchart to represent a storage area or queue.T/F
False. A rectangle is conventionally used in a process flowchart to represent a storage area or queue. Triangles are typically used in flowcharts to represent decision points, where a choice must be made between two or more alternatives.
Rectangles are used to represent activities or operations, while arrows connecting the shapes indicate the flow or direction of the process. The use of standardized shapes in flowcharts helps to make them easily understandable and accessible to a wide range of individuals, regardless of their background or experience with the specific process being depicted.
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Find the inverse of f(x):
f(x)=x^3+1
Step-by-step explanation:
y = x^3 + 1 switch x and y....then solve for y
x = y^3 + 1
x-1 = y^3
y = cubrt ( x-1) <====this is the inverse function
Can someone give me a step by step tut on how 135 divided by 20 is 6.75…I just forgot.
Answer: 135/20=6.75
Step-by-step explanation:
Long division (picture attached below)
Please help please and thank you its due this week on wen
Answer: 100
hope this helps
I NEED THE ANSWER TO THIS BY TODAY PLEASE HELP (200, 150) GCF
Answer: 50
Step-by-step explanation:
Greatest common factor of 200 and 150 is 50
200/50=4
150/50=3
a math teacher claimed that the average grade of the students in her algebra 2 classes this year would be equal to the average grade of the same students in algebra 1 classes two years ago. the average grade of algebra 1 students two years ago was 92%. in a random sample of 25 current algebra 2 students, the average grade was 87%, with a standard deviation of 7%. is there enough evidence to reject the teacher's claim?
From the hypothesis test conducted, the calculated t-statistic of -3.57 is less than the lower critical t-value of -2.064. This tells us that the observed sample mean of 87% is significantly different from the claimed population mean of 92% at the 0.05 significance level. Therefore, the null hypothesis can be rejected and conclusions can be made that that there is enough evidence to suggest that the average grade of the students in the Algebra 2 class this year is not equal to the average grade of the same students in the Algebra 1 class two years ago.
How do we conduct a hypothesis test?To determine whether there is enough evidence to reject the teacher's claim, we can conduct a hypothesis test. A one-sample t-test will be us to compare sample mean.
Here are the sample statistics:
Sample size (n) = 25
Sample mean (X) = 87%
Sample standard deviation (s) = 7%
A significance level (α) of 0.05, wil be used
We first calculate the t-statistic, with (X - μ) / (s/√n),
t = (87 - 92) / (7/√25) = -5 / (7/5) = -3.57
Then, we check this value against the critical t-value for a two-tailed test with 24 degrees of freedom (n-1), and α = 0.05.
The critical t-values for a two-tailed test at α = 0.05 and df = 24 are approximately -2.064 and +2.064.
-3.57 t-test is less than the lower critical t-value of -2.064. This means that the observed sample mean of 87% is significantly different from the claimed population mean of 92% at the 0.05 significance level.
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How do you do this. Please help right now.
The surface area of the prism is 520 in².
The area of the prism is 750 in³.
We have,
The given figure is a rectangular prism.
The surface area can be calculated by adding all the areas of the surfaces.
Now,
There are two types of surfaces.
Top and bottom surfaces and the side surfaces
So,
Top surface = Bottom surface
Area = 2 x (8 x 7(1/2)) = 8 x 15 = 120
Side surface.
Area = 4 x (8 x 12(1/2)) = 4 x (8 x 25/2) = 4 x 4 x 25 = 16 x 25 = 400
The surface area.
= 120 + 400
= 520 in²
And,
The area of the prism.
= 8 x 7(1/2) x 12(1/2)
= 8 x 15/2 x 25/2
= 2 x 15 x 25
= 30 x 25
= 750 in³
Thus,
The surface area of the prism is 520 in².
The area of the prism is 750 in³.
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suppose a 95% confidence interval for obtained from a random sample of size 13 is (3.5990, 19.0736). find the sample variance (round off to the nearest integer).
Therefore, the sample variance is approximately 65 using 95% confidence interval.
To find the sample variance, we need to use the formula for the confidence interval of the mean:
X ± tα/2 (s/√n)
Where:
X is the sample mean
tα/2 is the critical value for the t-distribution at the desired confidence level (with n-1 degrees of freedom)
s is the sample standard deviation
n is the sample size
From the given information, we know that the sample size n = 13, and the confidence interval is (3.5990, 19.0736) with a 95% confidence level. This means that the critical value for the t-distribution at α/2 = 0.025 and 11 degrees of freedom is approximately 2.201.
Using the formula for the confidence interval and the upper and lower bounds of the interval, we can solve for s:
19.0736 = X + 2.201(s/√13)
3.5990 = X - 2.201(s/√13)
Adding the two equations gives:
22.6726 = 2X
Therefore, X = 11.3363
Substituting this value of X into one of the original equations gives:
19.0736 = 11.3363 + 2.201(s/√13)
Solving for s, we get:
s ≈ 8.0587
Rounding to the nearest integer, the sample variance is:
s^2 ≈ 65
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What is the value of x in the given figure?
40°
210
O 20
O 40
O 50
O
85
Using the lines in the diagram shown, how many pentagons (of any size) can you trace?
Each line in every set of lines of the same color intersects with each line in two other sets. Because of that fact, there's a segment formed on each line in every set. Therefore, by tracing any segment from every set, we get a closed shape, which is a pentagon, as a result. Since there are 3 segments in each set and a pentagon has 5 sides, the number of pentagons we can trace is [tex]3\cdot3\cdot3\cdot3\cdot3=3^5=243[/tex].
0.612 and the 12 is repeating as a fraction
The Decimal 0.612 and the 12 is repeating as a fraction is 20.346/33.
Step 1: Let x = 0.612
Step 2: Multiply both sides by 1000 to remove the decimal point: 1000x = 612
Step 3: Since the 12 is repeating, we can subtract the non-repeating part from the repeating part to get 12 - 0 = 12. Then we can write the repeating part as a fraction with a denominator of 99 (which is one less than 100, the number of repeating digits). So we have: 0.12 = 12/99
Step 4: Rewrite the equation from step 2 using the value from step 3: 1000x = 612 + 12/99
Step 5: Simplify the right-hand side: 1000x = 61038/99
Step 6: Divide both sides by 1000 to solve for x: x = 61.038/99
Step 7: Simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 3: x = 20.346/33
Therefore, the decimal 0.612 and the 12 is repeating as a fraction is 20.346/33.
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according to census 2000, the highest proportion of married-couple households was in which state? question 14 options: utah mississippi montana
According to Census 2000, Utah had the highest proportion of married-couple households.
During the Census 2000, data was collected on the marital status of households in various states across the United States. The analysis revealed that Utah had the highest proportion of married-couple households among the given options of Utah, Mississippi, and Montana. This indicates that a larger percentage of households in Utah consisted of married couples compared to the other states during that time period. The Census data provides valuable insights into the demographic composition of different regions, helping to understand societal trends and patterns.
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hat is the surface area of this right rectangular prism with dimensions of 6 inches by 4 inches by 12 inches? A.324 B.256 C.288 D.512
The surface area of the given right rectangular prism is 288 square inches.
The given dimensions of the right rectangular prism are:
Length = 6 inches
Width = 4 inches
Height = 12 inches
The surface area of a rectangular prism can be calculated using the formula:
Surface Area = 2lw + 2lh + 2wh
Substituting the values into the formula:
Surface Area = 2(6)(4) + 2(6)(12) + 2(4)(12)
Surface Area = 48 + 144 + 96
Surface Area = 288 square inches
Therefore, the surface area of the given right rectangular prism is 288 square inches.
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Which of the following is true when using the empirical rule for a set of sample data? Select one: a. Approximately 68% of all observations are in the interval + s. b. Almost all observations are in the interval #2s. c. Approximately 68% of all observations are in the interval + 2s. d. Approximately 95% of all observations are in the interval + S.
When using the empirical rule for a set of sample data, the following statements are true: Approximately 68% of all observations are in the interval ± 1s (one standard deviation from the mean),Almost all observations are in the interval ± 3s (three standard deviations from the mean),Approximately 95% of all observations are in the interval ± 2s (two standard deviations from the mean),This statement, "Approximately 95% of all observations are in the interval + S", is incorrect.
When using the empirical rule for a set of sample data, the following statements are true:
a. Approximately 68% of all observations are in the interval ± 1s (one standard deviation from the mean).
b. Almost all observations are in the interval ± 3s (three standard deviations from the mean).
c. Approximately 95% of all observations are in the interval ± 2s (two standard deviations from the mean).
d. This statement, "Approximately 95% of all observations are in the interval + S", is incorrect. The empirical rule states that approximately 68% of all observations are in the interval ± 1s, not just "+ s".
Therefore, the correct answer is (c) Approximately 68% of all observations are in the interval + 2s.
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Find the y-intercept of the line on the graph.
(0, 0) is the y-intercept of the given line.
To detect the y-intercept of a line in a graph, you need to find the point where the line intersects the y-axis. In other words, it's the point where the value of x is zero.
To find the y-intercept of a line on a graph:
Locate the point where the line intersects the y-axis.Identify the x-coordinate of that point, which is always zero.The y-coordinate of that point is the y-intercept of the line.If the equation of the line is in slope-intercept form, y = mx + b, where m is the slope of the line and b is the y-intercept, then the value of b is the y-coordinate of the point where the line intersects the y-axis.
In the given graph line is paasing through origin (0, 0).
Thus, the value of y where the value of x will be zero is zero.
So the y-intercept is 0.
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a company has estimated that the probabilities of success for three products introduced in the market are 1/9, 2/3 and 1/5 respectively. assuming independence, find the probability that none of the products is successful?
The probability that none of the three products is successful is calculated by multiplying the probabilities of failure (1 - probability of success) for each product. Assuming independence, the calculation is as follows:
Product 1 failure probability: 1 - 1/9 = 8/9
Product 2 failure probability: 1 - 2/3 = 1/3
Product 3 failure probability: 1 - 1/5 = 4/5
Probability of all products failing: (8/9) × (1/3) × (4/5) = 32/135
Since the probabilities of success for each product are independent, we can simply multiply the probabilities of failure for each product to find the overall probability of none of the products being successful. This is a basic principle of probability when dealing with independent events.
The probability that none of the three products is successful in the market is 32/135.
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the equation of a line is y+7=3x+11 what is the value of y at the point where the line crosses the y-axis?
The value of y where it crosses the y-axis is (0, 4).
Given is an equation of line y+7 = 3x+11, we need to find the value of y where it crosses the y-axis,
So this mean we need to find the y-intercept,
To find the same put x = 0,
y+7 = 3(0) + 11
y+7 = 11
y = 4
Therefore, the y-intercept is (0, 4)
Hence the value of y where it crosses the y-axis is (0, 4).
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Which equation represents a circle that has a radius of 4 and a center at (4,-4)
Answer:The circle has a center at (5,5) and a radius of 2. Therefore, the equation is (x-5)2+(y-5)2=22, or (x-5)2+(y-5)2=4.
Step-by-step explanation:
i did the qestion on my test
A paper company produces 3,925 notebooks in 5 days. How many notebooks can it produce in 13 days?
A.10,150
B.10,205
C.2,041
D.3,753
B
The paper company can produce 785 papers in one day.
(3925÷5=785)
Therefore the paper company can produce 10205 papers in 13 days.
(785×13=10205)
Answer: B. 10,205
Step-by-step explanation: If in 5 days it produces 3,925 notebooks. Then what you should do is divide 3925 by 5. The answer for that equation is 785. So in 1 day they produce 785 notebooks. The last step you should do is multiply 785 by 13. 13 as in days and 785 as in the amount of notebooks made in one day. That's how you get 10,205 notebooks. Hope this helps :)
a small independent physicians practice has three doctors. dr. sarabia sees 27% of the patients, dr. tran sees 32% and dr.jackson sees the rest. dr.sarabia requests blood tests on 10% of her patients. dr. tran requests blood tests on 10% of his patients, and dr.jackson requests blood tests on 15% of his patients. an auditor randomly selects a patient from the past week. what is the probability that the patient had a blood test requested during his visit ? (use 3 decimals)? (use 3 decimals)
The probability that the randomly selected patient had a blood test requested during their visit is approximately 0.120
In this case:
Dr. Sarabia sees 27% of the patients and requests blood tests on 10% of them.Dr. Tran sees 32% of the patients and requests blood tests on 10% of them.Dr. Jackson sees the rest of the patients (100% - 27% - 32% = 41%) and requests blood tests on 15% of them.We can calculate the overall probability as the weighted average of the probabilities from each doctor:
Probability = (Probability from Dr. Sarabia) * (Percentage of patients seen by Dr. Sarabia) + (Probability from Dr. Tran) * (Percentage of patients seen by Dr. Tran) + (Probability from Dr. Jackson) * (Percentage of patients seen by Dr. Jackson)
Probability = (0.10 * 0.27) + (0.10 * 0.32) + (0.15 * 0.41)
Probability ≈ 0.027 + 0.032 + 0.061
Probability ≈ 0.12
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5. Ben solved the right triangle below. His work is shown below: 6^2 + 4^2 = 36 + 16 = 52. Then he took the square root of 52 and he got: 7.2. Analyze what Ben did wrong by writing a statement: What Ben did wrong was ___
The triangle is 6, 4 and x
The solution is: 5 in hours did Ben work that Saturday.
The number of hours that Ben worked at the restaurant can be calculated by the expression, Xpm - Xam.
The number of hours that Ben worked can be represented as:
= Time he finished working - Time he started working
= X pm - X am
If for instance, he started work at 7 am and finished at 12 pm, he would have worked:
= 12 - 7
= 5 hours
The expression that shows the length of time he worked can therefore be concluded to Xpm - Xam.
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complete question:
Ben is a waiter at a restaurant. on Saturday, Ben got up at 6:30am, started work at X.am and finished work at X.pm. how long in hours did Ben work that Saturday
Find the lateral area of a regular prism with height h if the base of the prism is a square with side 2cm and h=125mm
The lateral surface area of a regular prism would be =100cm²
How to calculate the lateral surface area of a prism?To calculate the lateral surface area of a prism the formula that should be used would be given below such as follows;
Lateral surface area of prism = perimeter of base×height
The base = perimeter of square = 4a
where;
a = side length = 2cm × 4 = 8cm
Height = 125 mm = 12.5cm
Lateral surface area of prism = 8×12.5 = 100cm²
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|x/3| if x< 0 I need help
Step-by-step explanation:
If x < 0, we consider the absolute value of x/3.
The absolute value is always positive or zero, which means when x < 0, the expression becomes -x/3. This is because we remove the negative sign from negative numbers when finding the absolute value.
So, |x/3| = (-x)/3 when x < 0.
Answer:
-x/3
Step-by-step explanation:
because x < 0 , so lets take x = -3
Then | -3/3 |
= | -1 | = -(-1) = 1
There fore , | x/3 | for x < 0 = -x/3
two sides of a triangle have the following measures. find the range of possible measures forthe third side. 1.) 9, 5, ___ >x> ___ 2.) 5,8, ___ >x> ___ 3.) 6, 10, ___ >x> ___ 4.) 6, 9, ___>x> ___ 5.) 11, 8, ___ >x> ___
1.) For the triangle with sides 9 and 5, the third side must satisfy the inequality:
9 - 5 < x < 9 + 5
which simplifies to:
4 < x < 14
So the range of possible measures for the third side is 4 < x < 14.
2.) For the triangle with sides 5 and 8, the third side must satisfy the inequality:
8 - 5 < x < 8 + 5
which simplifies to:
3 < x < 13
So the range of possible measures for the third side is 3 < x < 13.
3.) For the triangle with sides 6 and 10, the third side must satisfy the inequality:
10 - 6 < x < 10 + 6
which simplifies to:
4 < x < 16
So the range of possible measures for the third side is 4 < x < 16.
4.) For the triangle with sides 6 and 9, the third side must satisfy the inequality:
9 - 6 < x < 9 + 6
which simplifies to:
3 < x < 15
So the range of possible measures for the third side is 3 < x < 15.
5.) For the triangle with sides 11 and 8, the third side must satisfy the inequality:
11 - 8 < x < 11 + 8
which simplifies to:
3 < x < 19
So the range of possible measures for the third side is 3 < x < 19.
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Find the difference between the smallest and largest 4 digit numbers
whose values does not change on reversing the digits
Using number theory, the difference between the smallest and largest is 8998.
What is the difference between the smallest and largest 4 digit numbers whose values does not change on reversing the digits?In order to determine the difference between the largest and smallest 4-digit in which the values do not change when their digits are reversed, we can start with;
This problem falls under number theory and we can apply some insight here;
In this case, the smallest value will be 1000 while the largest will be 9999
We can add 1 from 1000 till we reach 9999. For each number, we compare it with its reversed form. If they are equal, we store that number as a candidate for the smallest and largest numbers.
After checking all the numbers, we find that the smallest and largest numbers that meet the given criterion are:
Smallest number: 1001 (remains the same when digits are reversed)
Largest number: 9999 (remains the same when digits are reversed)
Now, we can calculate the difference between the smallest and largest numbers:
Difference = Largest number - Smallest number
Difference = 9999 - 1001
Difference = 8998
Therefore, the difference between the smallest and largest 4-digit numbers that do not change when their digits are reversed is 8998.
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On the first day, a man walked 3 /8 of a journey. On the next day he covered another 2/5 of the journey. what fraction of the journey was covered?
Answer:
31/40
Step-by-step explanation:
3/8+2/5
Find the common denominator.
15/40+16/40
Add.
=31/40
Answer: 31/40
Step-by-step explanation:
3/8+2/5
make denominators equal, multiply by 8and 5
15/40+16/40=31/40
Which graph represents the equation (x – 3)2 + y2 = 16? On a coordinate plane, a circle has a center at (negative 3, 0) and radius 4. On a coordinate plane, a circle has a center at (negative 3, 0) and radius 16. On a coordinate plane, a circle has a center at (3, 0) and radius 4. On a coordinate plane, a circle has a center at (3, 0) and radius 16.
The statement that represents the equation (x – 3)2 + y2 = 16 is On a coordinate plane, a circle has a center at (3, 0) and radius 4.
The given equation is (x – 3)^2 + y^2 = 16
We solve to obtain:
(x – 3)^2 + (y - 0)^2 = 16
(x – 3)^2 + (y - 0)^2 = 4^2
This is an equation of a circle
Therefore, we can conclude that On a coordinate plane, a circle has a center at (3, 0) and radius 4.
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Three pieces of wood are cut from a plank 1 metre long.
Each piece is 30 cm long.
How long is the piece left over
Answer:
The piece left over is 10 cm long
Step-by-step explanation:
1 metre = 100 cm
Each piece = 30 cm long
Number of piece = 3
30 × 3 = 90
100 - 90 = 10 cm
Therefore, the piece left over is 10 cm long
