If we choose the point (-4, -2) in quadrant III and the point (3, -2) in quadrant IV on the same horizontal line, then the distance between these two points is 7 units.
In quadrant III, the x-coordinate and y-coordinate of a point are both negative, while in quadrant IV, the x-coordinate is positive, and the y-coordinate is negative.
If we choose a point in quadrant III and a point in quadrant IV on the same horizontal line, it means that they have the same y-coordinate, but their x-coordinates have opposite signs.
To find the distance between these two points using absolute value, we can first find the difference between their x-coordinates, ignoring their signs. We then take the absolute value of this difference to get the distance between them.
For example, let's say we choose the point (-4, -2) in quadrant III and the point (3, -2) in quadrant IV on the same horizontal line.
Their y-coordinates are the same (-2), so we can ignore them.
The difference between their x-coordinates is (-4) - 3 = -7.
To get the distance between them, we take the absolute value of this difference, which is | -7 | = 7.
Therefore, the distance between these two points is 7 units.
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A random sample of 100 automobile owners shows that in the state ofVirginia, an automobile is driven on the average 23,500 kilometersper year with a standard deviation of 3900 kilometers.
b) What can we assert with 99% confidence about the possible sizeof error if we estimate the average number of kilometers driven bycar owners in Virginia is to be 23500 per year?
Based on the information provided, we can assert with 99% confidence that the possible size of error in estimating the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year is within a range of +/- 774.14 kilometers. This is calculated using the formula:
Margin of error = z-score x (standard deviation / square root of sample size)
Where the z-score for a 99% confidence level is 2.576. Plugging in the values, we get:
Margin of error = 2.576 x (3900 / square root of 100)
Margin of error = 2.576 x 390
Margin of error = 1004.64 / 2
Margin of error = +/- 774.14
Therefore, we can assert that the estimated average number of kilometers driven by car owners in Virginia is within a range of 22,725.86 kilometers to 23,274.14 kilometers with 99% confidence.
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Identify why this assignment of probabilities cannot be legitimate: P(A) = 0.4, P(B) = 0.3, and P( AB=0.5 (A) A and B are not given as disjoint events (B) A and B are given as independent events (GP(A and B) cannot be greater than either P(A) or P(B) (D) The assignment is legitimate
The assignment of probabilities cannot be legitimate because of option (C): P(A and B) cannot be greater than either P(A) or P(B). In this case, P(AB) = 0.5, which is greater than both P(A) = 0.4 and P(B) = 0.3. For probabilities to be valid, the intersection of two events (A and B) must not exceed the individual probabilities of each event.
The reason why this assignment of probabilities cannot be legitimate is because of option B - A and B are given as independent events, but option A - A and B are not given as disjoint events. If A and B are independent events, then the probability of their intersection (AB) should be equal to the product of their individual probabilities, which is not the case here (0.5 ≠ 0.4 x 0.3). Therefore, option D - The assignment is legitimate is incorrect. Additionally, option C - GP(A and B) cannot be greater than either P(A) or P(B) is also violated, but it is not the main reason why the assignment is illegitimate.
The assignment of probabilities cannot be legitimate because of option (C): P(A and B) cannot be greater than either P(A) or P(B). In this case, P(AB) = 0.5, which is greater than both P(A) = 0.4 and P(B) = 0.3. For probabilities to be valid, the intersection of two events (A and B) must not exceed the individual probabilities of each event.
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find the point on the curve r(t)=(2cost 2sint e^t)
To find the point on the curve r(t)=(2cost, 2sint, e^t) we need to evaluate the x, y, and z coordinates at a specific value of t. Let's choose t=0 for simplicity.
So, plugging in t=0 we have:
r(0) = (2cos(0), 2sin(0), e^0)
r(0) = (2, 0, 1)
Therefore, the point on the curve at t=0 is (2, 0, 1).
To find the point on the curve r(t) = (2cos(t), 2sin(t), e^t) for a specific value of t, you can plug in the value of t into the parameterized equation:
1. Replace t with the specific value you're interested in (e.g., t = a).
2. Compute 2cos(a) for the x-coordinate.
3. Compute 2sin(a) for the y-coordinate.
4. Compute e^a for the z-coordinate.
The point on the curve will be (2cos(a), 2sin(a), e^a).
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A store ships books by weight.
A small box can hold 4−6
pounds.
A large box can hold 8−9
pounds.
The weights of the books are given below.
Drag books into each box to show what it can hold.
We can drag the books to the correct box paying attention to the number that represents their weight and how much each box can hold, as seen below.
The small box can hold:
1 blue book + 1 green book2 green booksThe large box can hold:
1 blue book and 2 green booksAdding the weightAccording to the attached picture, the small box can hold between 4 and 6 pounds. Therefore, we will only add to it the number of books whose weight falls between those two numbers. If they weigh less than 4 or more than 6, we shall not add them to the box. Thus:
1 blue book + 1 green book = 3.1 + 2.4 = 5.5 lb
2 green books = 2.4 + 2.4 = 4.8 lb
The same reasoning goes for the large box. It can hold between 8 and 9 pounds. Thus, we will not add anything under 8 or over 9.
1 blue book and 2 green books = 3.1 + 2.4 + 2.4 = 7.9 lb
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In each of the following find the pdf of Y.
(a) Y = X2 and fX(x) = 1, 0 < x < 1
(b) Y = −log X and X has pdf
The PDF of Y is1/(2*sqrt(y)), for 0 < y < 1 and fX(e^(-Y)) * |-e^(-Y)|.
(a) To find the pdf of Y when Y = X² and fX(x) = 1, 0 < x < 1, we first find the inverse function of Y = X², which is X = sqrt(Y). Next, we compute the derivative of X with respect to Y: dX/dY = 1/(2*sqrt(Y)). Now, we apply the change of variables formula to find the pdf of Y:
fY(y) = fX(x) * |dX/dY| = 1 * |1/(2*sqrt(y))| = 1/(2*sqrt(y)), for 0 < y < 1.
(b) To find the pdf of Y when Y = -log(X) and X has pdf, we first find the inverse function of Y = -log(X), which is X = e^(-Y). Next, we compute the derivative of X with respect to Y: dX/dY = -e^(-Y). We need the pdf of X to proceed, which is not provided in the question. Assuming we have the pdf of X as fX(x), we apply the change of variables formula to find the pdf of Y:
fY(y) = fX(x) * |dX/dY| = fX(e^(-Y)) * |-e^(-Y)|.
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PLEASE HELP ASAP !!!!
I put the scale model as well you ´ ll need it
It should be noted that to accurately calculate the area of the larger house, one must initialise by ascertaining the scale factor of the original.
How to explain the informationThe customer and constructor hold antithetical perspectives, hence influencing their appraisals. The former presumes that the linear expansion of the area is direct proportionate to the scale factor; thus implying that an augmentation in upscaling would lead to an equal rise in size.
Relying on this, the more enlarged property possesses an area four times greater than the original since, when augmenting the earlier measurements by two, the resultant increase in acreage is marked as (2 x 2 = 4). Accordingly, the total region of the extended residence becomes quadruple the acreage of the initial home.
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You are buying makeup for $4. 30. There is a 40% off sale on all makeup. The tax rate is 6%
The price paid for buying makeup of $4.30 after a discount of 40% and the tax rate of 6% is $2.84
Marked price = $4.30
Discount percent = 40%
Discount price = Marked price - Discount amount
Discount amount = 40% of 4.30
= 0.4 * 4.30 = $ 1.72
Discount price = 4.30 - 1.72
= $ 2.58
Tax = 6%
Tax = 0.06 * 4.30
= $ 0.258
Final price = Discount price + tax
= 2.58 + 0.258
Final price = $2.84
The money paid for buying the makeup after a 40% discount and 6% tax is $2.84.
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The complete question can be "You are buying makeup for $4. 30. There is a 40% off sale on all makeup. The tax rate is 6%. What is the final price that you pay?"
Jace wrote a sentence as an equation.
56 is 14 more than a number.
14 + p = 56
Which statement best describes Jace’s work?
Jace is not correct. The phrase more than suggests using the symbol > and Jace did not use that symbol.
Jace is not correct. He was correct to use addition, but the equation should be 56 + p = 14.
Jace is not correct. The first number in the sentence is 56, so the equation should start with 56.
Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56.
The required, Option D "Jace is correct. The phrase more than suggests addition, so Jace showed that 14 plus a variable equals 56." is correct.
Jace is correct. The sentence "56 is 14 more than a number" implies that you can start with a certain number (which is unknown) and add 14 to it to get 56. Jace correctly used addition to represent this relationship in equation 14 + p = 56, where p represents the unknown number. The phrase "more than" does not necessarily suggest the use of the symbol >, as it can also be interpreted as an additional relationship.
Therefore, Jace's work is accurate and correctly represents the relationship between 56 and a certain number that is 14 less than 56.
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Which expression is equivalent to (65. 85)3?
Answer: 197.55
Step-by-step explanation:
multiply the two numbers 65.85 times 3
pls hlp me I need it
The required function that has a similar rate of change as the function y = 7/4x + 3 is shown in C. Option C is correct.
Here,
Given function y = 7/4x + 3
Comparing the above equation with the standard equation of the linear function is y = mx + c
So, m = 7/4, which implies the slope is positive,
Similarly, the slope of the relationship given in the options is,
A, m = -1
B, m = 8/3
C = 7/4
D = -2
Thus, the required function that has a similar rate of change as the function y = 7/4x + 3 is shown in C. Option C is correct.
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A triangle has an angle that measures 110°. The other two angles are in a ratio of 3:11. What are the measures of those two angles?
The measure of the other two angles of the triangle in the ratio 3:11 are 15° and 55° respectively.
What is ratioA ratio is a comparison of two or more numbers that indicates their sizes in relation to each other. It can be used to express one quantity as a fraction of the other ones.
We recall that the sum of interior angles of a triangle is 180° and given one angle 110° and the ratio of the other two as 3: 11, then by comparison using the variable x;
110° + 3x + 11x = 180°
110° + 14x = 180°
14x = 180° - 110° {collect like terms}
14x = 70°
x = 70/14 {divide through by 14}
x = 5
so the two angles will be:
3(5) = 15°
11(5) = 55°
Therefore, the measure of the other two angles of the triangle in the ratio 3:11 are 15° and 55° respectively.
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You spin the spinner and flip a coin. Find the probability of the compound event.
The probability of spinning an even number and flipping heads is .
The compound event probability of spinning an even number and flipping heads is P ( A ) = 1/4
Given data ,
Let the total number of outcomes in spinning a spinner be = 6
The number of even outcomes = 3
Let the number of outcomes in flipping a coin = 2
So , the probability of landing an even number = 3/6 = 1/2
The probability of landing a heads = 1/2
And , probability of spinning an even number and flipping heads is given by P ( A ) = probability of landing a heads x probability of landing an even number
On simplifying , we get
P ( A ) = 1/2 ( 1/2 )
P ( A ) = 1/4
Hence , the probability is 1/4 or 25 %
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Complete the statement so that it is TRUE: The angle subtended by an arc at the centre of a circle is ...........
Answer:
equal to twice the angle subtended by the same arc at any point on the circumference of the circle. This is known as the central angle of the arc.
Step-by-step explanation:
chatgpt
bardAI
sam's probability of getting an a on an individual test is 75%. if he takes 12 tests, whatis the probability he gets as on exactly 10 of those tests?
The probability that Sam gets exactly 10 A's out of 12 tests is 0.234 or about 23.4%.
To solve this problem, we can use the binomial probability formula. The formula is:
[tex]P(X=k) = (n choose k) * p^k * (1-p)^(n-k)[/tex]
Where:
- P(X=k) is the probability of getting exactly k successes
- n is the number of trials (tests)
- k is the number of successes (getting an A)
- p is the probability of success on each trial (getting an A)
- (n choose k) is the binomial coefficient, which can be calculated as [tex]n! /[/tex][tex](k! * (n-k)!)[/tex]
Using this formula, we can plug in the values given in the problem:
n = 12 (number of tests)
k = 10 (number of A's)
p = 0.75 (probability of getting an A)
P(X=10) = (12 choose 10) * 0.75^10 * (1-0.75)^(12-10)
P(X=10) = (12! / (10! * 2!)) * 0.75^10 * 0.25^2
P(X=10) = 66 * 0.0563 * 0.0625
P(X=10) = 0.234
Therefore, the probability that Sam gets exactly 10 A's out of 12 tests is 0.234 or about 23.4%.
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for each of the figures write an absolute value equation that has the following solution set
ASAP PLS
The absolute value equation that has a solution set of -1/2 and 3/2 is [tex]|x - \frac{1}{2} | = -1[/tex].
We are given the solution sets of the absolute value equation and we have to find the absolute value equation. The solution sets given are
x = { -1/2, 3/2}. The absolute value of the equation can be represented as [tex]| x -x_{1} | - x_{2} = 0[/tex] in which [tex]x_{1}[/tex] is the mean of the given two numbers in the set and [tex]x_{2}[/tex] is the difference between the two numbers divided by 2.
Firstly, we will find the mean of the two numbers
[tex]x_{1}[/tex] = (-1/2 + 3/2)/ 2
[tex]x_{1}[/tex] = 1/2
Now, we will find the value of [tex]x_{2}[/tex]
[tex]x_{2}[/tex] = (-1/2 -3/2)/2
[tex]x_{2}[/tex] = -2/2
[tex]x_{2}[/tex] = -1
We will substitute the values of [tex]x_{1}[/tex] and [tex]x_{2}[/tex] in the absolute value equation.
[tex]| x -x_{1} | - x_{2} = 0[/tex]
[tex]| x - 1/2| - (-1) = 0[/tex]
[tex]| x - \frac{1}{2}| + 1 = 0[/tex]
[tex]| x - \frac{1}{2}| = -1[/tex]
Therefore, the absolute value equation that has a solution set of -1/2 and 3/2 is [tex]|x - \frac{1}{2} | = -1[/tex].
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find x refer to attached image
The requried measure of the tangent IP is evaluated as x = 11.
For the given circle, following the property of tangent and secant to the circle,
Tangent² = secant × IN
Substitute the value in the above expression,
x² = 20×IN
x² = 20 * 6
x = √120
x ≈ 11
Thus, the requried measure of the IP is evaluated as x = 11.
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Based on the simulations from parts (A) thru (E), what is the shape of the distribution of "heads"? (This is a reading assessment question. Be certain of your answer because you only get one attempt on this question.) Choose the correct shape of the distribution below. A. Skewed left B. Bell-shaped C. Skewed right
Based on the simulations from parts (A) thru (E), the shape of the distribution of "heads" is bell-shaped. Therefore, the correct answer is B. Bell-shaped.
What is the most probable distribution's shape?
Bell-shaped distributions, sometimes referred to as normal or Gaussian distributions in math and science, are the most significant probability distribution shapes since they are typically the result of sufficiently big data sets from naturally occurring random variables.
The mathematical idea known as the normal distribution, and also referred to as the Gaussian distribution, is commonly defined by the mound form.
When data points are utilized to plot a line for a particular item that satisfies the requirements of the normal distribution, a form called a mound is produced.
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What is n Rounded to the nearest 10
Answer:
9.9735
because 9.9735 can we rounded to 10 instead being confused about it
1. for a data set f x( ) to be modeled by an exponential function, which relationship is necessary between successive values of f x( ) for equally spaced values of x? a the first differences are approximately equal. b the second differences are approximately equal. c the ratios are approximately equal.
To model a data set f(x) by an exponential function for equally spaced values of x, the necessary relationship between successive values of f(x) is:
c) The ratios are approximately equal.
For a data set f x( ) to be modeled by an exponential function, the necessary relationship between successive values of f x( ) for equally spaced values of x is that the ratios are approximately equal. This means that the ratio of any two consecutive values of f x( ) is approximately the same, regardless of the specific values of x chosen. This property is known as exponential growth or decay, and it is a fundamental characteristic of many natural and social phenomena. Therefore, if we observe that the ratios of successive values of f x( ) are approximately equal, we can conclude that the data set is likely to be modeled by an exponential function.
To model a data set f(x) by an exponential function for equally spaced values of x, the necessary relationship between successive values of f(x) is:
c) The ratios are approximately equal.
In other words, when the values of x are equally spaced, the ratio of f(x+1)/f(x) should be approximately the same for each successive pair of points in the data set. This constant ratio is a characteristic of exponential functions.
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2) Write MN as the sum of unit vectors for M(- 3/4, 5, 2/3) and N(6, - 9, ⅗)
The sum of unit vectors for M(- 3/4, 5, 2/3) and N(6, - 9, ⅗) is (27/4√(8329/900))i - (14/√(8329/900))j - (1/15√(8329/900))k.
To write MN as the sum of unit vectors, we first need to find the vector MN, which is the difference between the position vectors of M and N. Then, we divide this vector by its magnitude to obtain a unit vector in the same direction. Finally, we express MN as the sum of these unit vectors.
MN = N - M = (6 - (-3/4), -9 - 5, 3/5 - 2/3) = (27/4, -14, -1/15)
|MN| = √((27/4)² + (-14)² + (-1/15)²) = √(8329/900)
Unit vector in the direction of MN = MN/|MN| = (27/4√(8329/900), -14/√(8329/900), -1/15√(8329/900))
Therefore, MN can be written as the sum of unit vectors:
MN = (27/4√(8329/900))i - (14/√(8329/900))j - (1/15√(8329/900))k
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How many miles can i walk in 110 minutes if i walk 3 miles in 70 minutes?
The number of miles that one can walk in 110 minutes with such a pace that they cover 3 miles in 70 minutes is 4.72.
The unitary method refers to the process in which by dividing we find the value of one by dividing. And then finding the value of a specific quantity by multiplying the one by the given number.
Therefore, we are given the number of miles traveled in 70 minutes and we can find the number of miles covered in 1 minute by division.
Number of miles in 70 minutes = 3 miles
Number of miles in 1 minute = 3 ÷ 70
= 0.042 miles
According to the question, we have to find miles covered in 110 minutes. To find it we multiply the miles in one minute by 110.
Number of miles in 110 minutes = 0.042 * 110
= 4.72 miles
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00.110.220.0530.65find the mean of this probability distribution. round your answer to one decimal place.
The mean of this probability distribution is 2.3 when rounded to one decimal place.
finding mean:To find the mean of this probability distribution, you'll need to multiply each value by its corresponding probability and then sum the products.
1. Multiply each value by its probability:
- 0 * 0.1 = 0
- 1 * 0.2 = 0.2
- 2 * 0.05 = 0.1
- 3 * 0.65 = 1.95
2. Sum the products:
- 0 + 0.2 + 0.1 + 1.95 = 2.25
So, the mean of this probability distribution is 2.3 when rounded to one decimal place.
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a white number cube, a red number cube, and a green number cube, each with faces numbered 1 to 6, are tossed at the same time. what is the probability of all three cubes landing on odd numbers given that the white cube landed on three?
the probability of all three cubes landing on odd numbers given that the white cube landed on three is 1/24
Since the white cube landed on 3, we only need to consider the red and green cubes.
The probability of each cube landing on an odd number is 1/2, since there are 3 odd numbers out of 6 total on each cube.
The probability of both the red and green cubes landing on odd numbers is the product of their individual probabilities:
P(red and green both odd) = P(red odd) x P(green odd) = 1/2 x 1/2 = 1/4
Therefore, the probability of all three cubes landing on odd numbers given that the white cube landed on three is:
P(white=3 and red and green both odd) = P(red and green both odd | white=3) x P(white=3)
= (1/4) x (1/6) = 1/24
So the probability of all three cubes landing on odd numbers given that the white cube landed on three is 1/24
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The p-value is _____ or less if the chi-square statistic is 6.63 or more.
The p-value is 0.01 or less if the chi-square statistic is 6.63 or more.
The p-value is 0.01 or less if the chi-square statistic is 6.63 or more.
An analytical statistical technique for categorical data is the chi-square test. A set of variables expected and observed frequencies can be compared to see if there is a significant difference.
The chi-square statistic, which gauges the difference between observed and anticipated frequencies, serves as the foundation for the chi-square test. The test looks at whether there is a substantial difference between the observed and expected data.
Numerous disciplines, including biology, psychology, sociology, and business, can benefit from the chi-square test. Frequently, it is used to test theories relating to the interactions between two or more category variables.
There are various chi-square tests, including the likelihood ratio chi-square test and Pearson's chi-square test. The chi-square test by Pearson
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Ayana measured a house and its lot and made a scale drawing. She used the scale 14 millimeters : 5 meters. In the drawing, the back patio is 84 millimeters long. What is the length of the actual patio?
The actual length of the back patio is 30 meters.
Given information:
Ayana measured a house and its lot and made a scale drawing. She used the scale 14 millimeters: 5 meters.
In the drawing, the back patio is 84 millimeters long.
If 14 millimeters in the drawing represent 5 meters in reality, then we can set up a proportion to find the length of the actual patio:
14 mm : 5 m = 84 mm : x
Cross-multiplying, we get:
14 mm (x) = 5 m (84 mm)
To simplify, we have:
x = (5 m 84 mm) / 14 mm
x = 30 m
Therefore, x = 30 m.
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what is the distance between -27 and 30 on a number line
Answer:
57
Step-by-step explanation:
Get the absolute value of subtracting -27 from 30 (absolute value essentially meaning you make the answer positive no matter what). -27-30 = -57, and its absolute value is 57, which is the distance.
The absolute value is there because the distance can't be negative, as in you can't travel negative 2 miles to the grocery store. Hope this helps.
When there is a treatment or behavior for which researchers want to study risk, they often compare it to the _______ risk, which is the risk without the treatment or behavior.
When there is a treatment or behavior for which researchers want to study risk, they often compare it to the baseline risk, which is the risk without the treatment or behavior.
The baseline risk provides a reference point for evaluating the effect of the treatment or behavior on the outcome of interest.
For example, if researchers want to study the risk of a certain disease among smokers compared to non-smokers, the baseline risk would be the risk of the disease among non-smokers. By comparing the risk of the disease among smokers to the baseline risk among non-smokers, researchers can determine the extent to which smoking increases the risk of the disease.
Similarly, if researchers want to study the risk of a certain side effect associated with a medication, the baseline risk would be the risk of the side effect in the absence of the medication. By comparing the risk of the side effect among people taking the medication to the baseline risk among people not taking the medication, researchers can determine the extent to which the medication increases the risk of the side effect.
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pls help asap will give points
The area of the given composite figure is: 156 sq. units
How to find the area of the composite figure?The formula for the area of a rectangle is expressed as:
Area = L * W
Where:
W is width
L is Length
The formula for the area of a circle is:
Area = πr²
Thus:
Area of composite figure = 13 * 12 = 156 sq. units
This is because the semi circle added is exactly the same area with the one removed.
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I'm lazy no judging
Find the Volume of this solid
Answer:
Step-by-step explanation:
21x9
181
when a test for steroids is given to soccer players, 96% of the players taking steroids test positive and 12% of the players not taking steroids test positive. suppose that 5% of soccer players take steroids. what is the probability that a soccer player who tests positive takes steroids? (enter the value of the probability in decimal format and round the final answer to three decimal places.)
The probability is 0.296 Therefore, the probability that a soccer player who tests positive takes steroids is about 0.296 or 29.6%
To find the probability that a soccer player who tests positive takes steroids, we can use Bayes' theorem.
1. Define the events:
A: A player takes steroids.
B: A player tests positive for steroids.
2. We are given the following probabilities:
P(B|A) = 0.96 (probability of testing positive given that a player takes steroids)
P(B|A') = 0.12 (probability of testing positive given that a player doesn't take steroids)
P(A) = 0.05 (probability that a player takes steroids)
3. We need to find P(A|B), the probability that a player takes steroids given that they tested positive.
The probability that a soccer player who tests positive takes steroids can be calculated using Bayes' theorem:
P(Steroids | Positive) = P(Positive | Steroids) * P(Steroids) / P(Positive)
where P(Positive | Steroids) = 0.96 (96% of players taking steroids test positive), P(Positive | No Steroids) = 0.12 (12% of players not taking steroids test positive), and P(Steroids) = 0.05 (5% of soccer players take steroids).
4. Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
5. We first need to find P(B). We can do this using the law of total probability:
P(B) = P(B|A) * P(A) + P(B|A') * P(A')
6. We know P(A) and can calculate P(A') as the complement:
P(A') = 1 - P(A) = 1 - 0.05 = 0.95
7. Now, we can calculate P(B):
P(B) = 0.96 * 0.05 + 0.12 * 0.95 = 0.048 + 0.114 = 0.162
8. Finally, we can calculate P(A|B) using Bayes' theorem:
P(A|B) = (0.96 * 0.05) / 0.162 ≈ 0.2963
The probability that a soccer player who tests positive takes steroids is approximately 0.296, or 29.6%.
Learn more about Probability:
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