Answer:
see below
Step-by-step explanation:
Let x be the original price
x* discount rate = discount
x * 60% = 30
Change to decimal form
x * .60 = 30
Divide each side by .60
x = 30/.60
x =50
The original price was 50 dollars
Answer:
A-30 B-20 C-50
Step-by-step explanation:
Two balls are drawn in succession out of a box containing 2 red and 5 white balls. Find the probability that at least 1 ball was red, given that the first ball was (Upper A )Replaced before the second draw. (Upper B )Not replaced before the second draw.
Answer:
With replacement = 14/49without replacement = 3/7Step-by-step explanation:
Since there are 2 red and 5 white balls in the box, the total number of balls in the bag = 2+5 = 7balls.
Probability that at least 1 ball was red, given that the first ball was replaced before the second can be calculated as shown;
Since at least 1 ball picked at random, was red, this means the selection can either be a red ball first then a white ball or two red balls.
Probability of selecting a red ball first then a white ball with replacement = (2/7*5/7) = 10/49
Probability of selecting two red balls with replacement = 2/7*2/7 = 4/49
The probability that at least 1 ball was red given that the first ball was replaced before the second draw= 10/49+4/49 = 14/49
If the balls were not replaced before the second draw
Probability of selecting a red ball first then a white ball without replacement = (2/7*5/6) = 10/42 = 5/21
Probability of selecting two red balls without replacement = 2/7*2/6 = 4/42 = 2/21
The probability that at least 1 ball was red given that the first ball was not replaced before the second draw = 5/21+4/21 = 9/21 = 3/7
The probability that at least 1 ball was red, given that the first ball was replaced before the second draw is 28.5%; and the probability that at least 1 ball was red, given that the first ball was not replaced before the second draw is 22.5%.
Since two balls are drawn in succession out of a box containing 2 red and 5 white balls, to find the probability that at least 1 ball was red, given that the first ball was A) replaced before the second draw; and B) not replaced before the second draw; the following calculations must be performed:
2 + 5 = X7 = X
(2/7 + 2/7) / 2 = X (0.285 + 0.285) / 2 = X 0.285 = X
(2/7 + 1/6) / 2 = X (0.28 + 0.16) / 2 = X 0.451 / 2 = X 0.225 = X
Therefore, the probability that at least 1 ball was red, given that the first ball was replaced before the second draw is 28.5%; and the probability that at least 1 ball was red, given that the first ball was not replaced before the second draw is 22.5%.
Learn more about probability in https://brainly.com/question/14393430
The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 51 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.2 pounds, what is the probability that the sample mean will be each of the following? Appendix A Statistical Tables a. More than 61 pounds
Answer:
0.007
Step-by-step explanation:
We were told in the above question that a random sample of 51 households is monitored for one year to determine aluminum usage
Step 1
We would have to find the sample standard deviation.
We use the formula = σ/√n
σ = 12.2 pounds
n = number of house holds = 51
= 12.2/√51
Sample Standard deviation = 1.7083417025.
Step 2
We find the z score for when the sample mean is more than 61
z-score formula is z = (x-μ)/σ
where:
x = raw score = 61 pounds
μ = the population mean = 56.8 pounds
σ = the sample standard deviation = 1.7083417025
z = (x-μ)/σ
z = (61 - 56.8)/ 1.7083417025
z = 2.45852
Finding the Probability using the z score table
P(z = 2.45852) = 0.99302
P(x>61) = 1 - P(z = 2.45852) = 0.0069755
≈ 0.007
Therefore,the probability that the sample mean will be more than 61 pounds is 0.007
The dimensions of a closed rectangular box are measured as 96 cm, 58 cm, and 48 cm, respectively, with a possible error of 0.2 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.
Answer:
161.6 cm²Step-by-step explanation:
Surface Area of the rectangular box = 2(LW+LH+WH)
L is the length of the box
W is the width of the box
H is the height of the box
let dL, dW and dH be the possible error in the dimensions L, W and H respectively.
Since there is a possible error of 0.2cm in each dimension, then dL = dW = dH = 0.2cm
The surface Area of the rectangular box using the differentials is expressed as shown;
S = 2{(LdW+WdL)+(LdH+HdL)+(WdH+HdW)]
Also given L = 96cm W = 58cm and H = 48cm, on substituting this given values and the differential error, we will have;
S = 2{(96*0.2+58*0.2) + (96*0.2+48*0.2)+(58*0.2+48*0.2)}
S = 2{19.2+11.6+19.2+9.6+11.6+9.6}
S = 2(80.8)
S = 161.6 cm²
Hence, the surface area of the box is 161.6 cm²
find the value of k if x minus 2 is a factor of P of X that is X square + X + k
Answer:
k = -6
Step-by-step explanation:
hello
saying that (x-2) is a factor of [tex]x^2+x+k[/tex]
means that 2 is a zero of
[tex]x^2+x+k=0 \ so\\2^2+2+k=0\\<=> 4+2+k=0\\<=> 6+k =0\\<=> k = -6[/tex]
and we can verify as
[tex](x^2+x-6)=(x-2)(x+3)[/tex]
so it is all good
hope this helps
Consider circle T with radius 24 in. and θ = StartFraction 5 pi Over 6 EndFraction radians. Circle T is shown. Line segments S T and V T are radii with lengths of 24 inches. Angle S T V is theta. What is the length of minor arc SV?
Answer:
20π inStep-by-step explanation:
Length of an arc is expressed as [tex]L = \frac{\theta}{2\pi } * 2\pi r\\[/tex]. Given;
[tex]\theta = \frac{5\pi }{6} rad\\ radius = 24in\\[/tex]
The length of the minor arc SV is expressed as:
[tex]L = \frac{\frac{5\pi }{6} }{2\pi } * 2\pi (24)\\L = \frac{5\pi }{12\pi } * 48\pi \\L = \frac{5}{12} * 48\pi \\L = \frac{240\pi }{12} \\L = 20\pi \ in[/tex]
Hence, The length of the arc SV is 20π in
Answer:
20 pi
Step-by-step explanation:
Suppose 150 students are randomly sampled from a population of college students. Among sampled students, the average IQ score is 115 with a standard deviation of 10. What is the 99% confidence interval for the average IQ of college students? Possible Answers: 1) A) E =1.21 B) E = 1.25 C) E =2.52 D) E = 2.11 2) A) 112.48 < μ < 117.52 B) 113.79 < μ < 116.21 C) 112.9 < μ < 117.10 D) 113.75 < μ < 116.3
Answer:
99% confidence interval for the mean of college students
A) 112.48 < μ < 117.52
Step-by-step explanation:
step(i):-
Given sample size 'n' =150
mean of the sample = 115
Standard deviation of the sample = 10
99% confidence interval for the mean of college students are determined by
[tex](x^{-} -t_{0.01} \frac{S}{\sqrt{n} } , x^{-} + t_{0.01} \frac{S}{\sqrt{n} } )[/tex]
Step(ii):-
Degrees of freedom
ν = n-1 = 150-1 =149
t₁₄₉,₀.₀₁ = 2.8494
99% confidence interval for the mean of college students are determined by
[tex](115 -2.8494 \frac{10}{\sqrt{150} } , 115 + 2.8494\frac{10}{\sqrt{150} } )[/tex]
on calculation , we get
(115 - 2.326 , 115 +2.326 )
(112.67 , 117.326)
please please please please help i need to pass please
Answer:
D
Step-by-step explanation:
Solution:-
The standard sinusoidal waveform defined over the domain [ 0 , 2π ] is given as:
f ( x ) = sin ( w*x ± k ) ± b
Where,
w: The frequency of the cycle
k: The phase difference
b: The vertical shift of center line from origin
We are given that the function completes 2 cycles over the domain of [ 0 , 2π ]. The number of cycles of a sinusoidal wave is given by the frequency parameter ( w ).
We will plug in w = 2. No information is given regarding the phase difference ( k ) and the position of waveform from the origin. So we can set these parameters to zero. k = b = 0.
The resulting sinusoidal waveform can be expressed as:
f ( x ) = sin ( 2x ) ... Answer
Irvin buys a car for $21 comma 804. It depreciates 25% each year that he owns it. What is the depreciated value of the car after 1 yr? after 2 yr? The depreciated value of the car after 1 yr is $? The depreciated value of the car after 2 yr is $?
Answer:
The depreciated value of the car after 1 yr is $16,353
The depreciated value of the car after 2 yr is $12,264.75
Step-by-step explanation:
Given
purchase amount P= $21,804
rate of depreciation R= 25%
applying the formula for the car deprecation we have
[tex]A= P*(1-\frac{R}{100} )^n[/tex]
Where,
A is the value of the car after n years,
P is the purchase amount,
R is the percentage rate of depreciation per annum,
n is the number of years after the purchase.
1. The depreciated value of the car after 1 yr is
n=1
[tex]A= 21,804*(1-\frac{25}{100} )^1\\\\A= 21,804*(1-0.25 )^1\\\\A= 21,804*0.75\\\\A= 16353[/tex]
The depreciated value of the car after 1 yr is $16,353
2. The depreciated value of the car after 2 yr is
n=2
[tex]A= 21,804*(1-\frac{25}{100} )^2\\\\A= 21,804*(1-0.25 )^2\\\\A= 21,804*0.75^2\\\\A= 21,804*0.5625\\\\A= 12264.75[/tex]
The depreciated value of the car after 2 yr is $12,264.75
Factor completely 2x3y + 18xy - 10x2y - 90y. I need this done today in a few minutes.
Answer:
2y (x^2+9) ( x-5)
Step-by-step explanation:
2x^3y + 18xy - 10x^2y - 90y
Factor out the common factor of 2y
2y(x^3+9x-5x^2-45)
Then factor by grouping
2y(x^3+9x -5x^2-45)
Taking x from the first group and -5 from the second
2y( x (x^2+9) -5(x^2+9))
Now factor out (x^2+9)
2y (x^2+9) ( x-5)
An industrial psychologist conducted an experiment in which 40 employees that were identified as "chronically tardy" by their managers were divided into two groups of size 20. Group 1 participated in the new "It's Great to be Awake!" program, while Group 2 had their pay docked. The following data represent the number of minutes that employees in Group 1 were late for work after participating in the program.
Does the probability plot suggest that the sample was obtained from a population that is normally distributed? Provide TWO reasons for your classification.
Answer:
The probability plot of this distribution shows that it is approximately normally distributed..
Check explanation for the reasons.
Step-by-step explanation:
The complete question is attached to this solution provided.
From the cumulative probability plot for this question, we can see that the plot is almost linear with no points outside the band (the fat pencil test).
The cumulative probability plot for a normal distribution isn't normally linear. It's usually fairly S shaped. But, when the probability plot satisfies the fat pencil test, we can conclude that the distribution is approximately linear. This is the first proof that this distribution is approximately normal.
Also, the p-value for the plot was obtained to be 0.541.
For this question, we are trying to check the notmality of the distribution, hence, the null hypothesis would be that the distribution is normal and the alternative hypothesis would be that the distribution isn't normal.
The interpretation of p-valies is that
When the p-value is greater than the significance level, we fail to reject the null hypothesis (normal hypothesis) and but if the p-value is less than the significance level, we reject the null hypothesis (normal hypothesis).
For this distribution,
p-value = 0.541
Significance level = 0.05 (Evident from the plot)
Hence,
p-value > significance level
So, we fail to reject the null or normality hypothesis. Hence, we can conclude that this distribution is approximately normal.
Hope this Helps!!!
Find the indicated conditional probability
using the following two-way table:
P( Drive to school | Sophomore ) = [?]
Round to the nearest hundredth.
Answer:
0.07
Step-by-step explanation:
The number of sophmores is 2+25+3 = 30.
Of these sophmores, 2 drive to school.
So the probability that a student drives to school, given that they are a sophmore, is 2/30, or approximately 0.07.
Answer:
[tex]\large \boxed{0.07}[/tex]
Step-by-step explanation:
The usual question is, "What is the probability of A, given B?"
They are asking, "What is the probability that you are driving to school if you are a sophomore (rather than taking the bus or walking)?"
We must first complete your frequency table by calculating the totals for each row and column.
The table shows that there are 30 students, two of whom drive to school.
[tex]P = \dfrac{2}{30}= \mathbf{0.07}\\\\\text{The conditional probability is $\large \boxed{\mathbf{0.07}}$}[/tex]
Perform the operation 3/a^2+2/ab^2
Answer:
Step-by-step explanation:
Least common denominator = a²b²
[tex]\frac{3}{a^{2}}+\frac{2}{ab^{2}}=\frac{3*b^{2}}{a^{2}*b^{2}}+\frac{2*a}{ab^{2}*a}\\\\=\frac{3b^{2}}{a^{2}b^{2}}+\frac{2a}{a^{2}b^{2}}\\\\=\frac{3b^{2}+2a}{a^{2}b^{2}}[/tex]
Convert into the following unit into 30 cm into miter
Answer:
it we'll be 0.3
Step-by-step explanation:
trust me man I like to explain but it's long
Answer:
0.3 meter or 3/10 meter
Step-by-step explanation:
As there are 100cm in 1 meter and you want to find 30cm in terms of meters.
It will be as
100cm = 1 meter (rule/lax)
100/100 cm = 1/100 meter (divide both sides of equation with 100)
1 cm = 1/100 meter
1 *30 cm = (1/100)*30 meter (multiply both sides with 30)
30 cm = 30/100 meter
30/100 more shortly can be written as 3/10 meter or in decimals 0.3 meter.
my dad is designing a new garden. he has 21 feet of fencing to go around the garden. he wants the length of the garden to be 1 1/2 feet longer than the width. how wide should he make the garden?
Answer:
21=2w+2w+3 18=4w w=4.5
If 2x+9<32 then x could be
Answer:
x < 11.5
Step-by-step explanation:
2x + 9 < 32
(2x + 9) - 9 < 32 - 9
2x < 23
2x/2 < 23/2
x < 11.5
Answer:
x < 11 1/2
Step-by-step explanation:
2x+9<32
Subtract 9 from each side
2x+9-9 < 32-9
2x<23
Divide by 2
2x/2 <23/2
x < 11 1/2
X is any number less than 11 1/2
Calculate the volume of a rectangular prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm. (As before, you do not need to enter the units since they are provided to the right of the answer box.)
Answer:
85.932 cm³
Step-by-step explanation:
The volume of rectangular prism is obtained as the product of its length (l) by its width (w) and by its height (h):
[tex]V=l*w*h[/tex]
The volume of a prism with a length of 4.4 cm, a width of 3.1 cm, and a height of 6.3 cm is:
[tex]V=4.4*3.1*6.3\\V=85.932\ cm^3[/tex]
The volume of this prism is 85.932 cm³.
Find the remainder when f(x)=2x3−x2+x+1 is divided by 2x+1.
Step-by-step explanation:
it can be simply done by using remainder theorem.
Fake Question: Should Sekkrit be a moderator? (answer if you can) Real Question: Solve for x. [tex]x^2+3x=-2[/tex]
Answer:
x = -2 , -1
Step-by-step explanation:
Set the equation equal to 0. Add 2 to both sides:
x² + 3x = -2
x² + 3x (+2) = - 2 (+2)
x² + 3x + 2 = 0
Simplify. Find factors of x² and 2 that will give 3x when combined:
x² + 3x + 2 = 0
x 2
x 1
(x + 2)(x + 1) = 0
Set each parenthesis equal to 0. Isolate the variable, x. Note that what you do to one side of the equation, you do to the other.
(x + 2) = 0
x + 2 (-2) = 0 (-2)
x = 0 - 2
x = -2
(x + 1) = 0
x + 1 (-1) = 0 (-1)
x = 0 - 1
x = -1
x = -2 , -1
~
Answer:
x = -2 OR x = -1
Step-by-step explanation:
=> [tex]x^2+3x = -2[/tex]
Adding 2 to both sides
=> [tex]x^2+3x+2 = 0[/tex]
Using mid-term break formula
=> [tex]x^2+x+2x+2 = 0[/tex]
=> x(x+1)+2(x+1) = 0
=> (x+2)(x+1) = 0
Either:
x+2 = 0 OR x+1 = 0
x = -2 OR x = -1
P.S. Ummmm maybe...... Because he usually reports absurd answers! So, Won't it be better that he could directly delete it. And one more thing! He's Online 24/7!!!!!
By what percent will the fraction increase if its numerator is increased by 60% and denominator is decreased by 20% ?
Answer:
100%
Step-by-step explanation:
Start with x.
x = x/1
Increase the numerator by 60% to 1.6x.
Decrease the numerator by 20% to 0.8.
The new fraction is
1.6x/0.8
Do the division.
1.6x/0.8 = 2x
The fraction increased from x to 2x. It became double of what it was. From x to 2x, the increase is x. Since x was the original number x is 100%.
The increase is 100%.
Answer:
33%
Step-by-step explanation:
let fraction be x/y
numerator increased by 60%
=x+60%ofx
=8x
denominator increased by 20%
=y+20%of y
so the increased fraction is 4x/3y
let the fraction is increased by a%
then
x/y +a%of (x/y)=4x/3y
or, a%of(x/y)=x/3y
[tex]a\% = \frac{x}{3y} \times \frac{y}{x} [/tex]
therefore a=33
anda%=33%
The length of a rectangle is 5M more than twice the width and the area of the rectangle is 63M to find the dimension of the rectangle
Answer:
width = 4.5 m
length = 14 m
Step-by-step explanation:
okay so first you right down that L = 5 + 2w
then as you know that Area = length * width so you replace the length with 5 + 2w
so it's A = (5 +2w) * w = 63
then 2 w^2 + 5w - 63 =0
so we solve for w which equals 4.5 after that you solve for length : 5+ 2*4.5 = 14
¿Cuál serie numérica tiene como regla general Xn = 2n +1?
a. 3, 5, 7, 9
b. 2, 4, 5, 8
c. 4, 6, 8,10
d. 2, 3, 4, 5
Answer:
The series of numbers that correspond to the general rule of [tex]X_n=2n+1[/tex] is {3, 5, 7, 9}.
Step-by-step explanation:
We are given with the following series options below;
a. 3, 5, 7, 9
b. 2, 4, 5, 8
c. 4, 6, 8,10
d. 2, 3, 4, 5
And we have to identify what number series has a general rule as [tex]X_n=2n+1[/tex].
For this, we will put the values of n in the above expression and then will see which series is obtained as a result.
So, the given expression is ; [tex]X_n=2n+1[/tex]
If we put n = 1, then;
[tex]X_1=(2\times 1)+1[/tex]
[tex]X_1 = 2+1 = 3[/tex]
If we put n = 2, then;
[tex]X_2=(2\times 2)+1[/tex]
[tex]X_2 = 4+1 = 5[/tex]
If we put n = 3, then;
[tex]X_3=(2\times 3)+1[/tex]
[tex]X_3 = 6+1 = 7[/tex]
If we put n = 4, then;
[tex]X_4=(2\times 4)+1[/tex]
[tex]X_4 = 8+1 = 9[/tex]
Hence, the series of numbers that correspond to the general rule of [tex]X_n=2n+1[/tex] is {3, 5, 7, 9}.
heres a list of numbers 3 6 9 7 4 6 7 0 7 Find median,mean,range and mode
median=order them and find the middle=6
mean=add them all up and divide by the amount of numbers=(3+6+9+7+4+6+7+0 +7)/9=5.4
range= the difference between the smallest and largest number=9-3=6
mode= the one that appears the most= 7
The median, mean, range and mode will be 6, 5.4, 9 and 7.
The median is the number in the middle when arranged in an ascending order. The numbers will be:
0, 3, 4, 6, 6, 7, 7, 7, 9.
The median is 6.
The range is the difference between the highest and lowest number which is: = 9 - 0 = 9
The mode is the number that appears most which is 7.
The mean will be the average which will be:
= (0 + 3 + 4 + 6 + 6 + 7 + 7 + 7 + 9) / 9.
= 49/9
= 5.4
Read related link on:
https://brainly.com/question/9426296
Find the area of this parallelogram.
6 cm
11 cm
Step-by-step explanation:
given,
base( b) = 6cm
height (h)= 11cm
now, area of parallelogram (a)= b×h
or, a = 6cm ×11cm
therefore the area of parallelogram (p) is 66cm^2.
hope it helps...
Suppose the sequence StartSet a Subscript n Baseline EndSet is defined by the recurrence relation a Subscript n plus 1equalsnegative 2na Subscript n, for nequals1, 2, 3,..., where a1equals5. Write out the first five terms of the sequence.
Answer:
-10, 40, -240, 1,920 and -19, 200
Step-by-step explanation:
Given the recurrence relation of the sequence defined as aₙ₊₁ = -2naₙ for n = 1, 2, 3... where a₁ = 5, to get the first five terms of the sequence, we will find the values for when n = 1 to n =5.
when n= 1;
aₙ₊₁ = -2naₙ
a₁₊₁ = -2(1)a₁
a₂ = -2(1)(5)
a₂ = -10
when n = 2;
a₂₊₁ = -2(2)a₂
a₃ = -2(2)(-10)
a₃ = 40
when n = 3;
a₃₊₁ = -2(3)a₃
a₄ = -2(3)(40)
a₄ = -240
when n= 4;
a₄₊₁ = -2(4)a₄
a₅ = -2(4)(-240)
a₅ = 1,920
when n = 5;
a₅₊₁ = -2(5)a₅
a₆ = -2(5)(1920)
a₆ = -19,200
Hence, the first five terms of the sequence is -10, 40, -240, 1,920 and -19, 200
The area of a rectangular horse pasture is 268,500 square yards. The length of the pasture is 5 yards less than three times the width. What is the width of the pasture in yards? Do not include units in your answer. Please help right away! Thank you very much!
Answer: width = 300
Step-by-step explanation:
Area (A) = Length (L) x width (w)
Given: A = 268,500
L = 3w - 5
w = w
268,500 = (3w - 5) x (w)
268,500 = 3w² - 5w
0 = 3w² - 5w - 268,500
0 = (3w + 895) (w - 300)
0 = 3w + 895 0 = w - 300
-985/3 = w 300 = w
Since width cannot be negative, disregard w = -985/3
So the only valid answer is: w = 300
Which value of x makes 7+5(x-3)=227+5(x−3)=227, plus, 5, left parenthesis, x, minus, 3, right parenthesis, equals, 22 a true statement? Choose 1 answer:
Answer:
7 + 5(x - 3) = 22
5(x - 3) = 15
x - 3 = 3
x = 6
Answer:
x = 6
Step-by-step explanation:
Step 1: Distribute 5
7 + 5x - 15 = 22
Step 2: Combine like terms
5x - 8 = 22
Step 3: Add 8 to both sides
5x = 30
Step 4: Divide both sides by 5
x = 6
Please help! V^2 = 25/81
Answer:
C and D
Step-by-step explanation:
khan acedemy
An equation is formed when two equal expressions. The solutions to the given equation are A, B, and C.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
The solution of the given equation v²=25/81 can be found as shown below.
v²=25/81
Taking the square root of both sides of the equation,
√(v²) = √(25/81)
v = √(25/81)
v = √(5² / 9²)
v = ± 5/9
Hence, the solutions of the given equation are A, B, and C.
Learn more about Equation here:
https://brainly.com/question/2263981
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The ratio of the areas of two circles is 121/100. What is the ratio of the radii of the two circles
Answer:
11/10
Step-by-step explanation:
The area ratio is the square of the radius ratio (k):
(121/100) = k²
k = √(121/100) = 11/10
The ratio of radii is 11/10.
A fair die is rolled repeatedly. Calculate to at least two decimal places:__________
a) the chance that the first 6 appears before the tenth roll
b) the chance that the third 6 appears on the tenth roll
c) the chance of seeing three 6's among the first ten rolls given that there were six 6's among the first twenty roles.
d) the expected number of rolls until six 6's appear
e) the expected number of rolls until all six faces appear
Answer:
a. 0.34885
b. 0.04651
c. 0.02404
d. 36
e. 14.7, say 15 trials
Step-by-step explanation:
Q17070205
Note:
1. In order to be applicable to established probability distributions, each roll is considered a Bernouilli trial, i.e. has only two outcomes, success or failure, and are all independent of each other.
2. use R to find the probability values from the respective distributions.
a) the chance that the first 6 appears before the tenth roll
This means that a six appears exactly once between the first and the nineth roll.
Using binomial distribution, p=1/6, n=9, x=1
dbinom(1,9,1/6) = 0.34885
b) the chance that the third 6 appears on the tenth roll
This means exactly two six's appear between the first and 9th rolls, and the tenth roll is a six.
Again, we have a binomial distribution of p=1/6, n=9, x=2
p1 = dbinom(2,9,1/6) = 0.27908
The probability of the tenth roll being a 6 is, evidently, p2 = 1/6.
Thus the probability of both happening, by the multiplication rule, assuming independence
P(third on the tenth roll) = p1*p2 = 0.04651
c) the chance of seeing three 6's among the first ten rolls given that there were six 6's among the first twenty roles.
Again, using binomial distribution, probability of 3-6's in the first 10 rolls,
p1 = dbinom(3,10,1/6) = 0.15504
Probability of 3-6's in the NEXT 10 rolls
p1 = dbinom(3,10,1/6) = 0.15504
Probability of both happening (multiplication rule, assuming both events are independent)
= p1 * p1 = 0.02404
d) the expected number of rolls until six 6's appear
Using the negative binomial distribution, the expected number of failures before n=6 successes, with probability p = 1/6
= n(1-p)/p
Total number of rolls by adding n
= n(1-p)/p + n = n(1-p+p)/p = n/p = 6/(1/6) = 36
e) the expected number of rolls until all six faces appear
P1 = 6/6 because the firs trial (roll) can be any face with probability 1
P2 = 6/5 because the second trial for a different face has probability 5/6, so requires 6/5 trials
P3 = 6/4 ...
P4 = 6/3
P5 = 6/2
P6 = 6/1
So the total mean (expected) number of trials is 6/6+6/5+6/4+6/3+6/2+6/1 = 14.7, say 15 trials
Profit Function for Producing Thermometers The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of x units of thermometers each week is represented by the function below, where x ≥ 0. Find the interval where the profit function P is increasing and the interval where P is decreasing. (Enter your answer using interval notation.) P(x) = −0.004x2 + 6x − 5,000 Increasing: Decreasing:
Answer:
Increasing: [tex](0, 750)[/tex]
Decreasing: [tex](750, \infty)[/tex]
Step-by-step explanation:
Critical points:
The critical points of a function f(x) are the values of x for which:
[tex]f'(x) = 0[/tex]
For any value of x, if f'(x) > 0, the function is increasing. Otherwise, if f'(x) < 0, the function is decreasing.
The critical points help us find these intervals.
In this question:
[tex]P(x) = -0.004x^{2} + 6x - 5000[/tex]
So
[tex]P'(x) = -0.008x + 6[/tex]
Critical point:
[tex]P'(x) = 0[/tex]
[tex]-0.008x + 6 = 0[/tex]
[tex]0.008x = 6[/tex]
[tex]x = \frac{6}{0.008}[/tex]
[tex]x = 750[/tex]
We have two intervals:
(0, 750) and [tex](750, \infty)[/tex]
(0, 750)
Will find P'(x) when x = 1
[tex]P'(x) = -0.008x + 6 = -0.008*1 + 6 = 5.992[/tex]
Positive, so increasing.
Interval [tex](750, \infty)[/tex]
Will find P'(x) when x = 800
[tex]P'(x) = -0.008x + 6 = -0.008*800 + 6 = -0.4[/tex]
Negative, then decreasing.
Answer:
Increasing: [tex](0, 750)[/tex]
Decreasing: [tex](750, \infty)[/tex]