The probability that every even number appears at least once before the first occurrence of an odd number is [tex]\frac{1}{20}[/tex]
Take [tex]A_{k}[/tex] - the event that odd number appeared on the k-th throw, B - every even number appeared at least once.
Let’s find P(B|[tex]A_{k}[/tex]) . Note that this probability is 0 for k∈{1,2,3} as you need k≥4 to place all distinct even numbers before the odd one.
Now for k≥4 we need to use the formula of inclusions and exclusions: P(B¯|[tex]A_{k}[/tex])=P(C2+C4+C6|[tex]A_{k}[/tex])=
=P(C2|[tex]A_{k}[/tex])+P(C4|[tex]A_{k}[/tex])+P(C6|[tex]A_{k}[/tex])−P(C2C4|[tex]A_{k}[/tex])−P(C2C6|[tex]A_{k}[/tex])−P(C4C6|[tex]A_{k}[/tex]) where Ci is the event that the dice i is missing.
These probabilities are:
P(Ci/[tex]A_{k}[/tex])=[tex](\frac{2}{3} )^{k-1}[/tex]
P(CiCj|[tex]A_{k}[/tex])=[tex](\frac{1}{3} )^{k-1}[/tex]
So
P(B|[tex]A_{k}[/tex])=1–P(B¯|[tex]A_{k}[/tex])=1−3⋅[tex](\frac{2}{3} )^{k-1}[/tex]+3[tex](\frac{2}{3} )^{k-1}[/tex].= 1 − [tex]\frac{9}{2}[/tex].[tex](\frac{2}{3} )^{k}[/tex]+9[tex](\frac{1}{3} )^{k}[/tex]
Now as P([tex]A_{k}[/tex])= [tex]\frac{1}{2^{k} }[/tex]we conclude that
P(B) = ∞∑k=4P(B/[tex]A_{k}[/tex])P([tex]A_{k}[/tex])= ∞∑k=4([tex](\frac{1}{2} )^{k}[/tex]−[tex]\frac{9}{2}[/tex][tex](\frac{1}{3} )^{k}[/tex]+9.[tex](\frac{1}{6} )^{k}[/tex])=
=[tex]\frac{\frac{1}{16} }{\frac{1}{2} }[/tex]−[tex]\frac{\frac{1}{18} }{\frac{2}{3} }[/tex]+[tex]\frac{\frac{1}{144} }{\frac{5}{6} }[/tex] = [tex]\frac{1}{8}[/tex] -[tex]\frac{1}{12}[/tex] + [tex]\frac{1}{120}[/tex] =[tex]\frac{6}{120}[/tex] = [tex]\frac{1}{20}[/tex]
the probability that every even number appears at least once before the first occurrence of an odd number is [tex]\frac{1}{20}[/tex]
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Joseph removes 5/8 gallon of white paint from a can. Then he adds 5/8 gallon of blue paint to the can.
Write and then evaluate a subtraction expression to find the overall increase or decrease in the amount of paint in the can.
Enter the correct answers in the boxes.
Expression:
gallon(s)
Joseph removes 5/8 gallon of white paint from a can.The amount of paint in the can = - 2.8 gallons
Step-by-step explanation:
Let X be the starting amount of paint in the can, in gallons.
"Removes 5/8 gallon" can be written -(5/8)gal
"Adds 3/8 gallon" becomes +(3/8)gal
The result is : X - (5/8)gal + (3/8)gal = Remaining Paint (gallons)
Consolidating: Remaining (gallons) = X - (2/8)gal
If we want the overall increase/decrease, subtract the initial volume:
Initial - Remaining = Change in amount of paint in can
Change = (X-(2/8)gal) - X = - (2/8) gallons
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I have one simple question. I have attached it, can you provide me with a step by step explanation of your answer, and please show your work.
The required equation model is y = -(4.4/1009)x + 16, its slope is -(4.4/1009), and the y-intercept of the line of fit is 16.
The given table shows the amounts of storage left y 7 (in gigabytes) on a music-playing device when there are x songs on MTR the device.
As per the given table, take two points (0, 16) and (1009, 11.6)
The required equation will be as:
y - 16 = [(11.6 - 16)/(1009-0)](x-0)
y - 16 = -(4.4/1009)x
y = -(4.4/1009)x + 16
The above equation models the amount of storage left as a function of the number of songs.
Here slope is -(4.4/1009), and y-intercept is 16
Therefore, the required equation model is y = -(4.4/1009)x + 16, its slope is -(4.4/1009), and the y-intercept of the line of fit is 16.
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The estimate number of cars that can be produced (C in thousands) at a manufacturing plant over the course of time (x in years) can be modeled by the polynomial function
The average rate of change between year 1 and year 10 is given as follows:
4.3 thousand cars a year.
How to obtain the average rate of change of a function?The average rate of change of a function is obtained by the change in the output divided by the change in the input. Hence, over an interval [a,b], the rate is given as follows:
[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]
In this problem, the function is presented as follows:
C(x) = 0.15x³ - 3.28x² + 23.75x - 2.1.
The interval is:
[1,10], hence a = 1, b = 10.
At x = 1, the numeric value is found replacing each instance of x by 1, hence:
C(1) = 0.15(1)³ - 3.28(1)² + 23.75(1) - 2.1 = 18.52 thousand cars.
At x = 10, the numeric value is found replacing each instance of x by 10, hence:
C(10) = 0.15(10)³ - 3.28(10)² + 23.75(10) - 2.1 = 57.4 thousand cars.
Then the average rate of change is obtained as follows:
r = (57.4 - 18.52)/(10 - 1) = 4.3 thousand cars a year.
Missing InformationThe problem is given by the image shown at the end of the answer.
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y+8= -7(x-2) in slope intercept form
Answer:
[tex]y=-7x+6[/tex]
Step-by-step explanation:
1 ) Our given equation is [tex]Y\left(x\right)+8=-7\left(x-2\right)[/tex]
2 ) We can now graph this equation ( desmos is a good online graphing calculator ). The graph is shown in the attachment...
3 ) Now that the equation has been graphed we can see that the slope intercept form is [tex]y=-7x+6[/tex]
Hope this helps! :)
Find the x value that makes the lines parallel.
2x°
80°
Answer:
x = 40
Step-by-step explanation:
for m and n to be parallel then
2x and 80 are alternate angles and congruent , so
2x = 80 ( divide both sides by 2 )
x = 40
Ms. Gallo buys a 64-ounce of peanut butter for $6.33. What is the cost per ounce
the radius of a sphere is increasing at a rate of 5 mm/s. how fast is the volume increasing (in mm3/s) when the diameter is 80 mm? (round your answer to two decimal places.)
The rate of change of the volume of the sphere is 100,530.96 mm^3/s
How to calculate rate of change of the volume of the sphere?
Given,
radius = r = 80 / 2 = 40 mm
rate of change radius = [tex]\frac{dr}{dt}[/tex] = 5 mm/s
Formula to calculate volume of the sphere is
[tex]V = \frac{4}{3}\pi r^{3}[/tex]
Differentiating the volume formula and we get new formula to calculate rate of change of the volume.
[tex]\frac{dV}{dt} = \frac{4}{3}\pi 3r^{2}\frac{dr}{dt}[/tex]
Simplify to
[tex]\frac{dV}{dt} = 4\pi r^{2}\frac{dr}{dt}[/tex]
Now, we put the given value to formula and calculate it
[tex]\frac{dV}{dt} = 4\pi (40)^{2}(5)[/tex]
[tex]\frac{dV}{dt} = 100,530.96[/tex] mm^3/s
Thus, the rate of change of the volume of the sphere is 100,530.96 mm^3/s
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which postulate or theorem can be used to prove <RPQ = <RNM
Answer:
alternate interior angles theorem
Step-by-step explanation:
alternate interior angles theorem states that, if 2 parallel lines and a transversal form alternate interior angles, then, said angles are congruent.
g(x)=-2x-1 find g(-1)
find the measure of each angle indicated
Answer:127 degrees
Step-by-step explanation: 180-74 = 106 106+21 = 127 180-127 = 53 180-53 =127
can someone please help me with problem, thank u.
Answer:
D
Step-by-step explanation:
First, we should find the equations of the two lines. The red line is just horizontal, and represents y = 4 while the blue line intersects the y-axis at -4 and every time it goes right one, it goes up 4, so it has a slope of four(also it looks kinda steep, so it cannot have a slope of 1/4). This means the equation is y = 4x - 4.
Notice the red line is dashed, that means the inequality is either less than or greater than. Also, the shaded part is below the vertical line, enclosing all the values of y less than 4, so the first inequality would be y < 4.
The blue line is solid, so it is either greater than or equal to or less than or equal to. To find out which one, we check to se where the point (0, 0) fits in:
y ? 4x - 4
0 ? 0 - 4
0 ? -4
Only a greater than or equal to sign would fit in: 0 ≥ -4
This means the answer is:
y < 4
y ≥ 4x - 4
which is option D
Answer:
Step-by-step explanation:
1)
The equation of a straight line:
y = k·x + b
x = 0
b = y = - 4
k = (y - y₀) / (x - x₀) = ( 4 - (-4)) / (2 - 0) = 8 / 2 = 4
y = 4·x - 4
2)
The equation of a straight line:
y = 4
3)
Shaded area:
y ≤ 4
y ≥ 4x - 4
Right answer:
D.
y ≤ 4
y ≥ 4x - 4
2f-3.7<2.3 inequality?
f < -5
> -1.3
≤ 5
≥ 1.3 pick one one each side
The solution to the inequality expression is f >-3
How to determine the solution to the inequality expression?From the question, we have the following parameter that can be used in our computation:
-2f - 3.7 < 2.3
Add 3.7 to both sides of the expression
This gives
-2f < 6
Divide both sides of the expression by -2
So, we have the following representation
f >-3
Hence, the solution is f >-3
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Calculate the area of a circular ring, whose internal and external radii are 3 cm and 10 cm respectively.
Step-by-step explanation:
area of circle = πr²
internal radii - 3 cm = r1
external radii - 10 cm = r2
internal + external radii = πr1² + πr2²
= 22/7×3² + 22/7×10²
= 28.28 + 314.28
= 342.56 cm²
The sum of the squares of two numbers is 32. the product of the two numbers is 16. find the numbers.
Answer:
4
Step-by-step explanation:
The answer really just popped into my head.
4^2 + 4^2 = 32
4x4=16
Let f and g be differentiable functions such that f(3)=5,g(3)=7,f
′
(3)=13,g
′
(3)=6,f
′
(7)=2, and g
′
(7)=0. If h(x)=(fog)(x), then h
′
(3)= ?? A. 14 B. 6 C. 12 D. 10
If h(x)=(fog)(x) then h'(3) = 12.
Given:
Let f and g be differentiable functions such that f(3)=5,g(3)=7,f'(3) = 13, g'(3) = 6,f'(7) = 2 and g'(7) = 0
h(x)=(fog)(x)
h(x) = f(g(x))
h'(x) = f'(g(x)) * g'(x)
h'(3) = f'(g(3) * g'(3)
= f'(7) * 6
= 2*6
= 12
Therefore If h(x)=(fog)(x) then h'(3) = 12.
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Write a linear function f with the values f(0) = −5 and f(1) =−3.
A function is f(x)=¯
Answer:
[tex]y=2x-5[/tex]
Step-by-step explanation:
We want to find a linear function [tex]f(x)[/tex] with the values
[tex]f(0)=-5\\f(1)=-3[/tex]
These mean that when x is 0, y is -5, and when x is 1, y is -3.
This means that we want to find the equation of a line that goes through the ordered pairs below:
[tex](0,-5)\\(1,-3)\\[/tex]
Let's find the slope between the two ordered pairs. The slope is the rise over the run. The rise is the change in y-values.
[tex]-3--5=-3+5=2[/tex]
The change in y-values is 2.
The run is the change in x values.
[tex]1-0=1[/tex]
The change in x-values is 1.
The slope is the rise divided by the run, so:
[tex]slope=\frac{2}{1}=2[/tex]
Now that we know the slope, let's find the y-intercept! This is when x equals 0. Luckily, that number is already given! We know that when x equals 0, y equals -5!
[tex]intercept=-5[/tex]
Now that we have the slope and intercept, we can plug these numbers into slope-intercept form!
[tex]y=2x-5[/tex]
The bookstore sold 40 biographies and
100 mysteries. Find the ratio of biographies
to mysteries sold.
Answer:
40;100
step by step explanation:
a rectangle is 6cm wide two different expressions are used to state it's length
Answer:
The length is 12
The area is 72[tex]cm^{2}[/tex]
Step-by-step explanation:
3x + 7 = 5x -17 Subtract 3x from both sides
7 = 2x -17 Add 17 to both sdies
24 = 2x Divide both sides by 2
12 = x
Area = length times witdth
Area = 6(12) = 72
Answer:
Step-by-step explanation:
part a):
6(3x+7)=6(5x-17)
part b):
18x+42=30x-102
42+102=30x-12x
144=12x
x=144/12
x=12
area = l x b
area = 6x{5(12)-17}
=6x(60-17)
=6x43
area=258cm2
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8-(-4f-3)≤ -f-8+2f
solve for f
Answer:
f ≤ - [tex]\frac{19}{3}[/tex]
Step-by-step explanation:
8 - (- 4f - 3 ) ≤ - f - 8 + 2f ← distribute parenthesis on left side by - 1
8 + 4f + 3 ≤ f - 8
4f + 11 ≤ f - 8 ( subtract f from both sides )
3f + 11 = - 8 ( subtract 11 from both sides )
3f ≤ - 19 ( divide both sides by 3 )
f ≤ - [tex]\frac{19}{3}[/tex]
Emma is x years old
Kate is 12 years younger than Emma. The sum of their ages is 24
Work out the age of Emma
Answer:
Step-by-step explanation:
Emma's age= x
Kate's age= x-12
sum of their ages=24
so Emma's age= 24-12
x=24-12
x=12
Emma's age=12
I need help please please
Value of angle T is 60°, angle U is 60°, TS is 13 cm and SU is 22.49 cm.
Given in question,
Triangle STU,
TU = 13 cm
∠S = 60°
ST = TU
= 13 cm
∠U = ∠S (Their sides are equal)
= 60°
∠S + ∠T + ∠U = 180°
60 + ∠T + 60 = 180
∠T + 120 = 180
∠T = 180 - 120
= 60
Hence, ∠T is 60°.
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5 divided by 1 and 2/3
Answer: The answer to that would be 3.
Step-by-step explanation: So 5 / 1 2/3 = 3
Sorry my handwriting is so sloppy! Hope it helps!
Please help 50 points
Step-by-step explanation:
look at where the dotted line is : it is vertical at x = -3.
that is automatically is equation : x = -3.
now, look at where the shaded side of that line is - to the left or to the right of the line ?
it is to the left of the line. and that message all values smaller than x = -3. but not equal indicated by the dotted line (that means the line itself is not included).
so, this is described by x < -3.
therefore, the first answer option is out.
now, look at the solid line.
what is the y-intercept (the y- value when x = 0) ?
when x = 0, the y-value is -1.
and everything to the left and below of the line is included (incl. the line itself, hence the solid line), so, the line inequality is then (just given by the available choices)
y <= -x - 1
and the 3rd answer option is therefore correct.
Answer:
[tex]\begin{cases}x < -3\\y \leq -x-1\end{cases}[/tex]
Step-by-step explanation:
When graphing inequalities:
< or > : dashed line.≤ or ≥ : solid line.< or ≤ : shade under the line.> or ≥ : shade above the line.From inspection of the given graph, there is a dashed line at x = -3 with shading under the line (to the left).
Therefore, the inequality that represents this is:
[tex]x < -3[/tex]The solid line has a negative slope. For each decrease of 1 unit in the y-direction, the x-values increase by 1 unit. Therefore, the slope of this line is -1.
From inspection of the given graph, the y-intercept of the solid line is -1.
Therefore, the equation of this line is y = -x - 1.
As the shading is under the line, the inequality that represents this is:
[tex]y \leq -x - 1[/tex].Therefore, the system of linear inequalities that represents the graph is:
[tex]\begin{cases}x < -3\\y \leq -x-1\end{cases}[/tex]
I know a) is 100
Confused about b) I got 81 but its wrong
Need an explanation along with answer
Step-by-step explanation:
a) 100
b)45
trust me it will helps you
Given h(x) = -2x - 5, solve for a when h(x) = 9.
X
Answer:
-23
Step-by-step explanation:
h(x) = -2x - 5
Let x = 9
h(9) = -2*9 - 5
h(9) = -18 - 5
= -23
Answer:
h(9) = -23
Step-by-step explanation:
Given explanation,
→ h(x) = -2x - 5
Then the value when h(x) = 9 is,
→ h(x) = -2x - 5
→ h(9) = -2(9) - 5
→ h(9) = -18 - 5
→ h(9) = -23
Hence, the value of h(9) is -23.
How do you write 6 hundredths as a decimal?
Answer:
.06
Step-by-step explanation:
6 hundredths
We want to write this as a decimal.
Writing decimals
This means write 6/100
There are two zeros so we need two numbers.
06/100
Then there are two numbers before the decimal.
We can write 06 before the decimal.
.06
6 hundredths can be written as 0.06 as a decimal.
What is fraction?
All fractions consist of a numerator and a denominator, separated by horizontal bars called fraction bars.
The denominator indicates the number of parts into which the whole is divided. Placed at the bottom of the break below the break bar. The
counter indicates how many sections of the fraction are displayed or selected. Placed at the top of the break above the break bar.
A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent.
6 hundredths means that when you divide "something" into a hundred equal parts, 6 hundredths are six of the newly divided parts.
When you divide 6 by hundreds, you get 6 hundredths as a decimal, which is 0.06.
Therefore, 6 hundredths can be written as 0.06 as a decimal.
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x + 25/-8 = -6 Solve for x.
Simplify your answer as much as possible.
Step-by-step explanation:
if there is nothing missing, we have
x + 25/-8 = -6
in order to compare or add or subtract fractions, we need to bring them all to the same denominator (bottom part).
remember, integer numbers are fractions too. like here
-6 = -6/1
25/-8 = -25/8
so, how can we bring -6/1 to .../8 ?
by multiplying 1 by 8.
but we cannot multiply only the denominator by 8. otherwise we would suddenly have
-6/8
and is -6/8 = -6/1 ? no, certainly not.
to keep the original value of the fraction we have to do the same multiplication also with the numerator (top part).
so, we actually do
-6/1 × 8/8 = -48/8
with this little trick we have now .../8 to operate with, and our transformed fraction has still the same value
-6/1 = -48/8 indeed.
so, we have
x + -25/8 = -48/8
x - 25/8 = -48/8
x = -48/8 + 25/8 = -23/8
you get a loan from the loan star bank to help pay for your home. find the interest on the loan if you borrowed $12,000 at 10% for one year.
The interest on the loan is $1200.
Given,
In the question:
You get a loan from the loan star bank to help pay for your home.
and, if you borrowed $12,000 at 10% for one year.
To find the interest on the loan.
Now, According to the question;
Based on the given conditions:
Formulate:
Borrowed= $12,000
At 10% for one year
12000 x 10%
Convert percentage into decimal
12000 x 0.1
calculate the product or quotient:
= 1200
Hence, The interest on the loan is $1200.
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A survey shows that 2/3 of all homeowners have a pet. Of those, 3/4 have either a dog or a cat. Of the homeowners, what fraction has either a dog or a cat ?
Answer:
Of the homeowners, half have either a dog or a cat.
Step-by-step explanation:
Since 2/3 of all homeowners have a pet, and 3/4 of those have dogs or cats, we simply need to multiply 2/3 and 3/4 to see what fraction of all homeowners have a dog or a cat.
2/3 x 3/4 = 6/12
6/12 = 1/2
We can double check this. If there are 60 homeowners, and 2/3 have a pet, that would mean that 40 of those homeowners have a pet.
If 3/4 of the 40 homeowners have either a dog or a cat, that would equate to 30 homeowners having either a dog or a cat.
Since 30 is half of 60, this would mean that 1/2 of the 60 homeowners have a dog or cat!
Hope this answer helped you!
[tex](8x^{6} )^{\frac{2}{3} }[/tex] someone explain?
Answer:
Step-by-step explanation:
so use the properties of powers, recall that powers like [tex]x^{2^{2} }[/tex] is the same as [tex]x^{2} *x^{2}[/tex] which can also be written as [tex]x^{2*2}[/tex] = [tex]x^{4}[/tex].
I want to make this clearer.
[tex]x^{3^{3} }[/tex] = [tex]x^{3}[/tex] * [tex]x^{3}[/tex] * [tex]x^{3}[/tex] = [tex]x^{3*3*3}[/tex] =[tex]x^{27}[/tex]
I'm not sure if that's making it clearer or not.
so for your question of [tex](8x^{6}) ^{2/3}[/tex] we can rewrite it to look like the above multiplication of powers
[tex]8x^{6*(2/3)}[/tex]
[tex]\frac{6}{1}[/tex] * [tex]\frac{2}{3}[/tex] = [tex]\frac{12}{3}[/tex] = 4
[tex]8x^{4}[/tex]
The first step in simplifying is to apply that outer exponent of [tex]\frac{2}{3}[/tex] to each factor inside the parentheses.
[tex]\left(8\cdot x^6\right)^{2/3} = \left(8\right)^{2/3} \cdot \left(x^6\right)^{2/3}[/tex]
Now using the concept that [tex](a)^{m/n} = \sqrt[m]{a^n}[/tex] for positive real numbers a and integers m and n, we can evaluate the first part:
[tex](8)^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} =4[/tex]
For the second part, we use the power-to-a-power rule, that says [tex](a^m)^n = a^{m\cdot n}[/tex]:
[tex]\left(x^6\right)^{2/3} = x^{6\cdot \frac{2}{3}} = x^4[/tex]
Putting that all together, we have
[tex]\left(8 x^6\right)^{2/3} = \left(8\right)^{2/3} \cdot \left(x^6\right)^{2/3} = 4x^4[/tex]