There are 91 ways to distribute the 12 identical computers among the elementary, middle, high school, and community college, ensuring that all 12 computers are given out.
To find the number of ways to distribute 12 identical computers among four schools (elementary, middle, high school, and community college) while ensuring that all 12 computers are given out, we can use the concept of stars and bars or the balls and urns method.
Let's represent the distribution using stars and bars. We have 12 identical stars (representing the computers) and 3 identical bars (representing the separators between the four schools). The bars divide the stars into four groups, each representing the number of computers given to each school.
We need to determine the number of ways to arrange the 12 stars and 3 bars. This can be calculated using the formula:
Number of ways = (n + k - 1) choose (k - 1) where n is the number of stars (12) and k is the number of bars (3).
Using this formula, the number of ways to distribute the 12 computers among the four schools is:
Number of ways = (12 + 3 - 1) choose (3 - 1)
= 14Choosee 2
= 91
Therefore, there are 91 ways to distribute the 12 identical computers among the elementary, middle, high school, and community colleges, ensuring that all 12 computers are given out.
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The time it takes for someone to finish a bowl of ramen can be modeled by a random variable with the following moment generating function:
M(t)= 1/ (1−0.05t)1,t<0.05
Find the variance of the time it takes for someone to finish a bowl of ramen.
Therefore, the variance of the time it takes for someone to finish a bowl of ramen is 4.6875.
Given, The moment generating function of the time it takes for someone to finish a bowl of ramen is
M(t)= 1/ (1−0.05t)1,t<0.05 We have to find the variance of the time it takes for someone to finish a bowl of ramen.
The variance of the random variable can be calculated by the formula Variance = M''(0) - [M'(0)]^2 where M(t) is the moment generating function of the random variable M'(t) is the first derivative of M(t)M''(t) is the second derivative of M(t)
We need to find M''(t) and M'(t)M(t) = 1/(1 - 0.05t)M'(t) = [0.05/(1 - 0.05t)^2]M''(t) = [0.1/(1 - 0.05t)^3] Now, at t = 0, M(0) = 1, M'(0) = 1.25, M''(0) = 6.25 Variance = M''(0) - [M'(0)]^2 Variance = 6.25 - (1.25)^2 Variance = 6.25 - 1.5625 Variance = 4.6875
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Given: The time it takes for someone to finish a bowl of ramen can be modeled by a random variable with the following moment generating function: M(t)= 1/ (1−0.05t)1,t<0.05. The variance of the time it takes for someone to finish a bowl of ramen is 400.
The moment generating function of a random variable is defined as [tex]$M(t) = \mathbb{E}(e^{tX})$[/tex] for all t in an open interval around 0 which X is a random variable.
We are given that the moment generating function of the random variable T is given by:
[tex]$$M(t)= \frac{1}{1-0.05t} ,\ t < 0.05$$[/tex]
The [tex]$n^{th}$[/tex] derivative of M(t) at 0 is given by:
[tex]$$\frac{d^n}{dt^n} M(t) \biggr|_{t=0} = \mathbb{E}(X^n)$$[/tex]
We differentiate $[tex]M(t)$[/tex] with respect to $t$ to get [tex]$$M'(t) = \frac{0.05}{(1 - 0.05t)^2}$$[/tex].
Differentiating [tex]$M'(t)$[/tex] with respect to [tex]$t$[/tex] we get [tex]$$M''(t) = \frac{2(0.05)^2}{(1-0.05t)^3}$$[/tex].
Differentiating [tex]$M''(t)$[/tex] with respect to [tex]$t$[/tex] we get [tex]$$M'''(t) = \frac{6(0.05)^3}{(1-0.05t)^4}$$[/tex].
Substituting t = 0, we get [tex]$$M'(0) = \frac{1}{0.05} = 20$$[/tex]
[tex]$$M''(0) = \frac{2}{(0.05)^3} = 800$$[/tex]
[tex]$$M'''(0) = \frac{6}{(0.05)^4} = 4800$$[/tex]
Using the following formula to calculate the variance of X: [tex]$$Var(X) = \mathbb{E}(X^2) - [\mathbb{E}(X)]^2$$[/tex], where [tex]$$\mathbb{E}(X^2) = M''(0) = 800$$[/tex].
[tex]$$[\mathbb{E}(X)]^2 = [M'(0)]^2 = 400$$[/tex]
Hence, we get:$$Var(X) = 800 - 400 = \boxed{400}$$.
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O There were 9 bags of
candy donated for the
neighborhood party.
Each bag contained
245 pieces. How much
candy did they have
for the party?
The highest temperature in Las Vegas is 125 degrees Fahrenheit and the lower recorded temperature in Las Vegas is 50 degrees Fahrenheit below zero what is the difference between these two temperatures
Answer:
175 degrees Fahrenheit
Step-by-step explanation:
We are to find the difference between the two temperatures
125 - (-50)
two minuses gives a plus
125 = 50 = 175
f(x) = 2x+10. If f(x)= -2, find x.
Answer:
x=-5
Step-by-step explanation:
A 12ft basketball hoop casts an 8 ft shadow. Find the length of the shadow of a 4 ft tall fence.
Set up a ratio of height over shadow for each :
12/8 = 4/x
Cross multiply:
12x = 32
Divide both sides by 12:
X = 2 2/3 feet
The shadow is 2 2/3 feet.
What is the range of the function shown on the graph above? The graph is in the photo
OA. -6 < y < 9
OB. -6 _< y _< 9
OC. 0 _< y _< 7
OD. 0 < y < 7
What is the measure of angle C?
Answer:
angle C = 36°
Step-by-step explanation:
Fun fact that I found out:
all interior angles of a triangle added together = 180°
5x + 3x + 2x = 180°
combine like terms:
10x = 180°
divide both sides of the equation by 10:
x = 18°
angle C = 2(18°) = 36°
If f is any function, then the associated Green's Function G[f] is given by G[f](x) = integral ^x_0 f(s) sin(x - s)ds. Use variation of parameters to show that G[f] is a solution of y" + y = f(x).
We have: u''(x) = ƒ(x)cot(x) - 2u'(x)cot(x).Thus, we can find a particular solution of this differential equation by using variation of parameters.
Let G(x) = ƒ(s)sin(x - s) ds.
Then, by the product rule, we have: G' = ƒ(s)cos(x - s) ds - ƒ(s)sin(x - s) ds, and G'' = -ƒ(s)sin(x - s) ds - ƒ(s)cos(x - s) ds. Hence, we have:G'' + G = ƒ(s)sin(x - s) ds - ƒ(s)cos(x - s) ds + ƒ(s)sin(x - s) ds = ƒ(s)sin(x - s) ds = G.
So, G is indeed a solution of y'' + y = ƒ(x).Next, we will use variation of parameters to find a second solution of the same differential equation.
Let us suppose that we have another solution of the form y = u(x) sin(x).
Then, y' = u(x)cos(x) + u'(x)sin(x), and y'' = - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x).
Substituting these into the differential equation, we get:- u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x) + u(x)sin(x) = ƒ(x)2u'(x)cos(x) + u''(x)sin(x) = ƒ(x)
Dividing by sin(x), we get:2u'(x)cot(x) + u''(x) = ƒ(x)cot(x).
Now, let us assume that the second solution is of the form y = u(x)sin(x), where u is a function to be determined.
Then, we have: y' = u(x)cos(x) + u'(x)sin(x) and y'' = - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x).
Substituting these into the differential equation, we get: - u(x)sin(x) + 2u'(x)cos(x) + u''(x)sin(x) + u(x)sin(x) = ƒ(x)2u'(x)cos(x) + u''(x)sin(x) = ƒ(x)
Dividing by sin(x), we get:2u'(x)cot(x) + u''(x) = ƒ(x)cot(x).
Hence, we have: u''(x) = ƒ(x)cot(x) - 2u'(x)cot(x).Thus, we can find a particular solution of this differential equation by using variation of parameters.
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25
What is the solution to the equation 12(x+5) = 4x?
Answer:
x = -7.5
Step-by-step explanation:
12(x+5) = 4x
12x+ 60 = 4x
60 = -8x
-7.5 = x
How many solutions does this equation have? 9z = –8 + 7z
-no solution
-one solution
-infinitely many solutions
Answer:
one solution.
Simplify the expression completely.
i have now attached the picture but it can be wrong!
help me, please. I'm not very good at math
Answer:
1st, 2nd, 3rd
Step-by-step 1explanation:
30+40+5=75
30+40=70
70+5=75
20+1+50+4
=20+50=70
1+4=5
70+5=75
50+30-5
50+30=80
80-5=75
I hope this helps :)
Find the point at which the line intersects the given plane. x = 2 - 2t, y = 3t, z = 1 + t: x + 2y - z = 7 (x, y, z) = Consider the following planes. 4x - 3y + z = 1, 3x + y - 4z = 4 (a) Find parametric equations for the line of intersection of the planes.
The parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4 are:
x = (208 + 70t) / 52
y = (13 + 19t) / 13
z = t
To find the parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4, we can solve these two equations simultaneously.
Step 1: Set up a system of equations:
4x - 3y + z = 1
3x + y - 4z = 4
Step 2: Solve the system of equations to find the values of x, y, and z. One way to solve the system is by using the method of elimination:
Multiply the first equation by 3 and the second equation by 4 to eliminate the y term:
12x - 9y + 3z = 3
12x + 4y - 16z = 16
Subtract the first equation from the second equation:
12x + 4y - 16z - (12x - 9y + 3z) = 16 - 3
12x + 4y - 16z - 12x + 9y - 3z = 13y - 19z = 13
Step 3: Express y and z in terms of a parameter, let's call it t:
13y - 19z = 13
y = (13 + 19z) / 13
We can take z as the parameter t:
z = t
Substituting the value of z in terms of t into the equation for y:
y = (13 + 19t) / 13
Step 4: Express x in terms of t:
From the first equation of the original system:
4x - 3y + z = 1
4x - 3((13 + 19t) / 13) + t = 1
4x - (39 + 57t) / 13 + t = 1
4x - (39 + 57t + 13t) / 13 = 1
4x - (39 + 70t) / 13 = 1
4x = (39 + 70t) / 13 + 1
x = ((39 + 70t) / 13 + 13) / 4
x = (39 + 70t + 169) / 52
x = (208 + 70t) / 52
Therefore, the parametric equations for the line of intersection of the planes 4x - 3y + z = 1 and 3x + y - 4z = 4 are:
x = (208 + 70t) / 52
y = (13 + 19t) / 13
z = t
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A continuous random variable is said to have a Laplace(μ, b) distribution if its probability density function is given by
fX(x)= 1 exp(−|x−μ|), 2b b
where μ is a real number and b>0.
(i). If X ∼ Laplace(0,1), find E(X) and Var(X).
(ii). If X ∼ Laplace(0,1) and Y = bX + μ, show Y ∼ Laplace(μ, b). (iii). If W ∼ Laplace(2,8), find E(W) and Var(W).
(i) For X ~ Laplace(0,1):
E(X) = 0, Var(X) = 2.
(ii) If X ~ Laplace(0,1) and Y = bX + μ:
Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8):
E(W) can be approximated numerically.
Var(W) = 128.
(i) If X ~ Laplace(0,1), we need to find the expected value (E(X)) and variance (Var(X)).
The Laplace(0,1) distribution has μ = 0 and b = 1. Substituting these values into the PDF, we have:
fX(x) = (1/2) * exp(-|x|)
To find E(X), we integrate x * fX(x) over the entire range of X:
E(X) = ∫x * fX(x) dx = ∫x * [(1/2) * exp(-|x|)] dx
Since the Laplace distribution is symmetric about the mean (μ = 0), the integral of an odd function over a symmetric range is zero. Therefore, E(X) = 0 for X ~ Laplace(0,1).
To find Var(X), we use the formula:
Var(X) = E(X^2) - [E(X)]^2
First, let's find E(X^2):
E(X^2) = ∫x^2 * fX(x) dx = ∫x^2 * [(1/2) * exp(-|x|)] dx
Using the symmetry of the Laplace distribution, we can simplify the integral:
E(X^2) = 2 * ∫x^2 * [(1/2) * exp(-x)] dx (integral from 0 to ∞)
Solving this integral, we get:
E(X^2) = 2
Now, substitute the values into the variance formula:
Var(X) = E(X^2) - [E(X)]^2 = 2 - 0 = 2
Therefore, for X ~ Laplace(0,1), E(X) = 0 and Var(X) = 2.
(ii) To show that Y = bX + μ follows a Laplace(μ, b) distribution, we need to find the probability density function (PDF) of Y.
Using the transformation method, let's express X in terms of Y:
X = (Y - μ)/b
Now, calculate the derivative of X with respect to Y:
dX/dY = 1/b
The absolute value of the derivative is |dX/dY| = 1/b.
To find the PDF of Y, substitute the expression for X and the derivative into the Laplace(0,1) PDF:
fY(y) = fX((y-μ)/b) * |dX/dY| = (1/2) * exp(-|(y-μ)/b|) * (1/b)
Simplifying this expression, we get:
fY(y) = 1/(2b) * exp(-|y-μ|/b)
This is the PDF of a Laplace(μ, b) distribution, thus showing that Y ~ Laplace(μ, b).
(iii) For W ~ Laplace(2,8), we need to find E(W) and Var(W).
The PDF of W is given by:
fW(w) = (1/16) * exp(-|w-2|/8)
To find E(W), we integrate w * fW(w) over the entire range of W:
E(W) = ∫w * fW(w) dw = ∫w * [(1/16) * exp(-|w-2|/8)] dw
This integral can be challenging to solve analytically. However, we can approximate the expected value using numerical methods or software.
To find Var(W), we can use the property that the variance of the Laplace distribution is given by 2b^2, where b is the scale parameter.
Var(W) = 2 * b^2
= 2 * (8^2)
= 2 * 64
= 128
Therefore, Var(W) = 128 for W ~ Laplace(2,8).
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AH PLEASE SOMEONE HELP
Answer:
Variant b
Step-by-step explanation:
If you multiply it should be a+b but if you divided
A+b-c
Help pls it is my homework
Can y'all help me?
Answer:
A
Step-by-step explanation:
the mean is what occurs most often
If You Have NO EXPLANATION Don't ANSWER
Answer:
B. A = 1/2(7)h
Step-by-step explanation:
Formula for area of triangle = 1/2 x base x height
H is the height of the triangle.
7cm is identified as the base of the triangle.
1/2(7)h is also the same thing as 1/2 x 7 x h basically.
Answer:
B
Step-by-step explanation:
The area (A) of a triangle is calculated as
A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the perpendicular height )
Here b = 7 and h = h , then
A = [tex]\frac{1}{2}[/tex] (7) h → B
y= 2x-3
y= x+4
Graph each system and determine the number of the solutions that it has. If it has one solution, name it.
PLEASE ASAP HELP!!!
Assume that the prevalence of breast cancer is 13%. The
diagnostic test has a sensitivity of 86.9% and a
specificity of 88.9%. If a patient gets a positive result
What is the probability that the patient has breast cancer?
The probability that the patient has breast cancer given a positive result is 62.2%.
The probability of testing positive given the patient has breast cancer is:
P(P|C) = 0.869
The specificity of the test is 88.9% or 0.889, meaning that the test will correctly identify 88.9% of patients who do not have breast cancer as not having the disease.
So, the probability of testing negative given the patient does not have breast cancer is:
P(N|N) = 0.889
Now, using Bayes' theorem:
P(C|P) = P(P|C) * P(C) / P(P)
where,P(P) = P(P|C) * P(C) + P(P|N) * P(N)
Here, P(P|N) is the probability of testing positive given that the patient does not have breast cancer. This is equal to 1 - specificity = 1 - 0.889 = 0.111.
So, P(P) = P(P|C) * P(C) + P(P|N) * P(N) = 0.869 * 0.13 + 0.111 * (1 - 0.13) = 0.1823
So,P(C|P) = 0.869 * 0.13 / 0.1823 = 0.622 or 62.2%
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In each case, write the principal part of the function at its isolated singular points and determine whether that point is a removable singular point, an essential singular point or a pole (please also determine the order m for a pole). Then calculate the residue of the corresponding singular point. a) ( nett for isolatod singular point = = -1 b) (x - 1)2022 exp(-) for isolated singular point = 1.
The principal part at the isolated singular point -1 is not provided, so we cannot determine its nature or residue. And b) The principal part at the isolated singular point 1 is (x - 1)^2022 exp(-1). It is a pole of order 2022, and its residue is 0.
a) The principal part at the isolated singular point -1 is not provided, so we cannot determine its nature (removable singular point, essential singular point, or pole) or calculate its residue without additional information.
b) The given function is (x - 1)^2022 exp(-1). At the isolated singular point x = 1, the principal part of the function is (x - 1)^2022 exp(-1). Here, (x - 1)^2022 represents the pole part of the function, and exp(-1) represents the non-pole part.
Since the term (x - 1)^2022 dominates near x = 1, we can conclude that x = 1 is a pole. The order of the pole is determined by the exponent of (x - 1), which is 2022 in this case.
To calculate the residue, we need more information about the function, specifically the coefficients of the Laurent series expansion near the singular point. Without that information, we cannot determine the residue at x = 1.
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Do males or females feel more tense or stressed out at work? A survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table shown to the right. Complete parts (a) through (d) below. Felt Tense or Stressed Out at Work Yes No Total Gender Male 100 200 300 Female 145 125 270 Total 245 325 570 a. What is the probability that a randomly selected person's gender is female?
b. What is the probability that a randomly selected person feels tense or stressed out at work and is female?
c. What is the probability that a randomly selected person feels tense or stressed out at work or is female?
d. Explain the difference in the results in (b) and (c).
A survey of employed adults conducted online by a company on behalf of a research organization revealed the data in the contingency table is as follows:
a) The probability that a randomly selected person's gender is female is 270/570 or 0.474, which is approximately 47.4%.Formula used: P (Female) = Number of Females/Total Number of Individuals
b) The probability that a randomly selected person feels tense or stressed out at work and is female is 145/570 or 0.254, which is approximately 25.4%. Formula used: P (Female and Tense) = Number of Females who are Tense/Total Number of Individuals
c) The probability that a randomly selected person feels tense or stressed out at work or is female is: P (Female or Tense) = P(Female) + P(Tense) - P(Female and Tense)P(Tense) = (245/570) or 0.43, which is approximately 43%P(Female or Tense) = 0.47 + 0.43 - 0.254 = 0.646, which is approximately 64.6%.
d) The distinction between the outcomes in (b) and (c) is that the former shows the likelihood of being female and tense at work, whereas the latter shows the likelihood of being female or tense at work.
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Shania is making lasagna. The recipe she uses calls for 2 1/3 cups of spaghetti sauce. If she doubles the recipe, how much spaghetti sauce will she need?
Answer:
4 2/3 cups
Step-by-step explanation:
A donut has a diameter of 7 in. What is the radius?
Answer:
The radius is 3.5 inches I think.
Step-by-step explanation:
Hope this helped Mark BRAINLIEST!!!
Answer:
3.5
Step-by-step explanation:
You would simply divide 7 inches by 2 because the radius is one-half the measure of the diameter.
The cost of renting a bicycle, y, for
x hours can be modeled by a linear
function. Renters pay a fixed insurance
fee of $12 plus an additional cost of $10
per hour, for a maximum of 6 hours.
What is the range of the function for this
situation?
F {22, 32, 42, 52, 62, 72}
G {1, 2, 3, 4, 5, 6}
H {12, 24, 36, 48, 60, 72}
J {22, 34, 46, 58, 70, 82}
Answer:
F
Step-by-step explanation:
1(10) + 12= 22
2(10) + 12= 32
etc.....
FILL in the blank:AB E M nxn (R) (i) det (A.B) = ____________ . (ii) A is invertible if and only if _____________ .
Answer:
For square matrices A and B of equal size, the determinant of a matrix product equals the product of their determinants: det (A.B) = det (A) det (B) 1. A is invertible if and only if its determinant is nonzero 1.
Step-by-step explanation:
One kilogram is approximately 2.2 pounds. Write a direct variation equation that relates x kilograms to y pounds.
Answer:
2.2y=1x or just x
Step-by-step explanation:
Answer: y=2.2x
Step-by-step explanation:
what is the price of a $600 bike 15% off
Answer: You will pay $510 for a item with original price of $600 when discounted 15%.
Let R be the binary relation defined on a set of all integers Z as follows: for all integers m and n, mRn m’ – n’ is divisible by 6. a) Is R an equivalence relation? Check the conditions. b) What is the equivalence class of -17?
Previous question
The required solutions are:
a) Yes, the relation R is an equivalence relation.
b)The equivalence class of -17 is {-17, -23, -29, -35, ...}.
a) In order to determine whether R is an equivalence relation or not, we need to check if it satisfies the following three conditions:
Reflexibility: For all integers m, mRm should hold. In the given case, if we take m=n, we have m-n=n-m=0, which is divisible by 6. So, we can see that the reflexibility is satisfied.Transitivity: For all integers m, n, and p, if mRn and nRp hold, then mRp should also hold. Assume mRn and nRp, which means m-n, and n-p are both divisible by 6. To check transitivity, we need to check if m - p is divisible by 6. By adding the two previous conditions, we have (m-n) + (n-p) = m-p, which is also divisible by 6. Therefore, transitivity is satisfied.Symmetry: For all integers m and n, if mRn holds, then nRm should also hold. If mRn, it means m-n is divisible by 6. In order to check the symmetry, we need to check if n - m is divisible by 6. We can use the fact that a-b = -(b-a), we can rewrite n - m as -(m - n), which is divisible by 6. So, we can say that symmetry is satisfied.We can see that the relation 'R' satisfies all the conditions ( reflexibility, symmetry, and transitivity), so R is an equivalence relation.
b) In order to find the equivalence class of -17, we need to find all integers that are related to -17 under the relation R.
We can rewrite the relation as mRn if and only if m' - n' = 6k for some integer k.
In this case, -17Rn if and only if (-17)' - n' = -17 - n = 6k for some integer k.
To find all integers n that satisfy this equation, we can rearrange it as n = -17 - 6k.
By putting in different values of k, we can find all the integers n that are in the equivalence class of -17.
For example, when k = 0, n = -17 - 6(0) = -17. So, -17 is in the equivalence class of -17.
We can also see that when k = 1, n = -17 - 6(1) = -23. So, -23 is also in the equivalence class of -17.
The equivalence class of -17 consists of all integers that can be obtained by subtracting multiples of 6 from -17. So, the equivalence class of -17 is {-17, -23, -29, -35, ...}.
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The following contingency table gives the results of a sample survey of South African male and female respondents with regard to their preferred fastfood outlet: PREFERRED FAST FOOD OUTLET Burger King McDonalds TOTAL 50 No. of Males No. of Females 20 130 100 120 TOTAL 270 150 110 140 400 0.1.1.1 What is the probablyf randomly selecting a respondent who is male and prefer Burger 01.12 What is the probably selecting a female respondent, even that the preferred fastfood out? 0.1.1.3 What is the probability of selecting a respondent who is female or who prefers McDonalds? 12) (2) Events X and Yare such that PC) = 0.20 and PCXUY) = 0.55. Given that Xand Yare independent and non-mutually taclusive, determine P(Y). Give your final answer as a percentage to two decimal places (5) 13 (2) 2.1.3.1 Helen is the manager of a Finance Department. She has fifteen (15) members of stuff working for her. She has to choose five (5) members of her staff for a research team. How many different teams can she select from the fifteen members of staff 2.1.22 There are twelve (12) teams in a basketball league. What is the probability of correctly predicting the top three teams at the end of the 3) season in the correct order?
Q1.1.1 The probability of randomly selecting a male respondent from the sample is 0.4. Q.1.1.2 The probability of randomly selecting a respondent who is female and prefers HP is 0.275. Q.1.1.3 The probability of selecting a male respondent, given that the preferred brand is Lenovo is 0.4545. Q.1.1.4 The probability of selecting a respondent who is male or who prefers HP is 0.575. Q.1.1.5 The probability of selecting a respondent who does not prefer Lenovo is 0.725.
Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?
The probability of randomly selecting a male respondent is given by the number of male respondents divided by the total number of respondents:
Probability = No. of Males / Total = 160 / 400 = 0.4
Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?
The probability of randomly selecting a respondent who is female and prefers HP is given by the number of females who prefer HP divided by the total number of respondents:
Probability = No. of Females who prefer HP / Total = 110 / 400 = 0.275
Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?
The probability of selecting a male respondent, given that the preferred brand is Lenovo, is given by the number of males who prefer Lenovo divided by the total number of respondents who prefer Lenovo:
Probability = No. of Males who prefer Lenovo / Total No. of respondents who prefer Lenovo = 50 / 110 = 0.4545
Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?
The probability of selecting a respondent who is male or who prefers HP is given by the sum of the probabilities of selecting a male respondent and selecting a respondent who prefers HP, minus the probability of selecting both (to avoid double counting):
Probability = (No. of Males / Total) + (No. of Females who prefer HP / Total) - (No. of Males who prefer HP / Total)
Probability = (160 / 400) + (110 / 400) - (40 / 400) = 0.4 + 0.275 - 0.1 = 0.575
Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?
The probability of selecting a respondent who does not prefer Lenovo is given by the number of respondents who do not prefer Lenovo divided by the total number of respondents:
Probability = (Total - No. of respondents who prefer Lenovo) / Total
Probability = (400 - 110) / 400 = 290 / 400 = 0.725
The complete question is:
The following contingency table gives the results of a sample survey of South African male and female respondents with regard to their preferred brand of notebook:
HP Lenovo Dell Total
No. of Females 110 60 70 240
No. of Males 40 50 70 160
Total 150 110 140 400
Q.1.1.1 What is the probability of randomly selecting a male respondent from the sample?
Q.1.1.2 What is the probability of randomly selecting a respondent who is female and prefers HP?
Q.1.1.3 What is the probability of selecting a male respondent, given that the preferred brand is Lenovo?
Q.1.1.4 What is the probability of selecting a respondent who is male or who prefers HP?
Q.1.1.5 What is the probability of selecting a respondent who does not prefer Lenovo?
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