a)The probability Space, S = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6), (2, 1) (2, 2) (2,3) (2,5) (2,4),(2,6), (3,1) (3,2) ,(3,3), (3,4) (3,5)(3,6), (4,1) (4,2) (4,3) (4,4) (4,5) (4,6), (5,1) (5,2) (5,3) (5,4) (5,5) (5,6),(6,1) (6,2) (6,3) (6,4) (6,5) (6,4), (6,5),(6,6)}
b) i) Probability that the sum of the rolls is even, P(A) = 0.5
ii)Probability that first roll is equal to the second, P(B) 0.166
iii) Probability that the first roll is larger than the second, P(C) = 0.4166.
We have a pair of dice are rolled, then the Sample space is same as probability space.
a) The probability space , S is a non-empty set contains all possible outcomes. The probability space is written as
S = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6), (2, 1) (2, 2) (2,3) (2,5) (2,4),(2,6), (3,1) (3,2) ,(3,3), (3,4) (3,5)(3,6), (4,1) (4,2) (4,3) (4,4) (4,5) (4,6), (5,1) (5,2) (5,3) (5,4) (5,5) (5,6),(6,1) (6,2) (6,3) (6,4) (6,5) (6,4), (6,5),(6,6)}
b) Total possible outcomes from rolling = 36
Let us consider the events,
A : sum of the rolls is even
B : first roll is equal to the second
C : first roll is larger than the second.
Now, favourable outcomes for event A = 18
i) Probability (Sum is even), P( A) = 18/36
= 0.5
Favourable outcomes for event B = 6 = { (1,1),(2,2),(3,3),(4,4), (4,1) (4,2) (4,3), (5,5),(6,6)}
ii) So, Probability that first roll is equal to the second, P(B) = 6/36 = 0.166
Favourable outcomes for event C = 15 = { (2,1),(3,2),(3,1),(4,1),(5,1) (5,2) (5,3) (5,4),(6,1) (6,2) (6,3) (6,4) (6,5) (6,4), (6,5)}
iii) So, probability that the first roll is larger than the second, P(C) = 15/36
= 0.4166.
Hence, we get all required probabilities.
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The probability of P(Sum is even)=0.5, P(the first roll is equal to the second)=[tex]\frac{1}{6}[/tex], P(the first roll is larger than the second)=0.4167.
A pair of dice are rolled
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur.
The probability formula is defined as the possibility of an event to happen is equal to the ratio of the number of favourable outcomes and the total number of outcomes.a) Sample space,
[tex]$$\begin{aligned}S= & \left\{\begin{array}{l}(1,1)(1,2)(1,3)(1,4)(1,5)(1,6) \\(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)\end{array}\right. \\& (3,1)(3,2)(3,3)(3,4)(3,5)(3,6) \\& (4,1)(4,2)(4,3)(4,4)(4,5)(4,6) \\& (5,1)(5,2)(5,3)(5,4)(5,5)(5,6) \\& (6,1)(6,2)(6,3)(6,4)(6,5)(6,2)\}\end{aligned}$$[/tex]
(b).
(i).The probability of each of the following events
Probability of event to happen
P(E) = Number of favourable outcomes/Total Number of outcomes⇒P(Sum is even)=[tex]\frac{18}{36} =\frac{1}{2}[/tex]=0.5
⇒P(Sum is even)=0.5
(ii). P(the first roll is equal to the second)=[tex]\frac{6}{36} =\frac{1}{6}[/tex]
⇒P(the first roll is equal to the second)=[tex]\frac{1}{6}[/tex]
(iii). P(the first roll is larger than the second)=[tex]\frac{15}{36}=0.4167[/tex]
⇒P(the first roll is larger than the second)=0.4167
Therefore, the P(Sum is even)=0.5, P(the first roll is equal to the second)=[tex]\frac{1}{6}[/tex], P(the first roll is larger than the second)=0.4167.
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when the subtracts 4 from both sides 1/2x=-1/2
Subtracting 4 from both sides of the equation ¹/₂x = -¹/₂ gives us; ¹/₂x - 4 = -¹/₂ - 4
How to use subtraction property of equality?We want to simply the expression which is;
¹/₂x = -¹/₂
Now, when we subtract 4 from both sides, it means we are using subtraction property of equality which states that subtracting the same value from both sides of an equation makes the equation still to be equal. Thus, we have;
¹/₂x - 4 = -¹/₂ - 4
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pls can solve this question . thanks
PLS HELP FAST THERE'S A TIMER WILL GIVE BRAINLIETS.
Which name accurately describes the figure shown below and why? A 4-sided figure with 2 parallel sides, 2 right angles, and 2 non-congruent sides. parallelogram, because the figure has two pairs of parallel sides parallelogram, because the figure has exactly one pair of parallel sides trapezoid, because the figure has two pairs of parallel sides trapezoid, because the figure has exactly one pair of parallel sides Mark this and return
A meteorologist found that the rainfall in Vindale during the first half of the month was 1/2 of an inch. At the end of the month, she found that the total rainfall for the month was 2/3 of an inch. How much did it rain in the second half of the month?
The quantity of rainfall that fell in the second half of the month is; ¹/₆ of an inch.
How to solve fraction word problems?We are given;
Rainfall during first half of the month = 1/2 inch
Total rainfall for the month = 2/3 inch
Now, to find the rainfall for the second half of the month, we just subtract the one for the first half from the one for the month to get;
Rainfall in second half of the month = ²/₃ - ¹/₂ = ¹/₆ inches
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Find the time required for an investment of 5000 dollars to grow to 6000 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Answer:
2.4536 years
Step-by-step explanation:
compound formula:
A=P*(1+r/n)^nt
A = 6000
P = 5000
r = rate
n = 4 times per year compounded
t = time in years
6000 = 5000(1 + .075/4)^4t
divide both sides by 5000
1.2 = (1.01875)^4t
log both sides
log(1.2) = 4t * log(1.01875)
solve for t
t = 2.4537 years
f(x)=x+3
g(x) = 2x² - 4
Find (f. g)(x).
Answer:
[tex](f \cdot g)(x)[/tex] = [tex]2x^2 -1[/tex]
Step-by-step explanation:
[tex](f \cdot g)(x)[/tex], read as "f of g of x", means we have to use the function [tex]\bf g(x)[/tex] as input for the function [tex]\bf f(x)[/tex]. This means, we have to replace the [tex]x[/tex] in the definition of [tex]f(x)[/tex] with [tex]g(x)[/tex].
[tex]f(x)=x+3[/tex]
[tex]g(x) = 2x^2 - 4[/tex]
∴ [tex](f \cdot g)(x) = f(g(x))[/tex]
⇒ [tex]g(x) + 3[/tex]
⇒ [tex](2x^2 - 4) + 3[/tex]
⇒ [tex]2x^2 -1[/tex]
A bank gets an average of 12 customers per hour. Assume the variable follows a Poisson distribution. Find the probability that there will be 4 or more customers at this bank in one hour.
Using the Poisson distribution, there is a 0.9978 = 99.78% probability that there will be 4 or more customers at this bank in one hour.
What is the Poisson distribution?In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
The parameters are:
x is the number of successese = 2.71828 is the Euler number[tex]\mu[/tex] is the mean in the given interval.A bank gets an average of 12 customers per hour, hence the mean is [tex]\mu = 12[/tex].
The probability that there will be 4 or more customers at this bank in one hour is:
[tex]P(X \geq 4) = 1 - P(X < 4)[/tex]
In which:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
Then:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-12}12^{0}}{(0)!} \approx 0[/tex]
[tex]P(X = 1) = \frac{e^{-12}12^{1}}{(1)!} \approx 0[/tex]
[tex]P(X = 2) = \frac{e^{-12}12^{2}}{(2)!} = 0.0004[/tex]
[tex]P(X = 3) = \frac{e^{-12}12^{3}}{(3)!} = 0.0018[/tex]
Then:
P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0 + 0 + 0.0004 + 0.0018 = 0.0022.
[tex]P(X \geq 4) = 1 - P(X < 4) = 1 - 0.0022 = 0.9978[/tex]
0.9978 = 99.78% probability that there will be 4 or more customers at this bank in one hour.
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A rectangle is 6 meters long and 4 meters wide. What is the area of the rectangle?
24 cm 2
10 cm 2
48 cm 2
20 cm 2
Please help I don’t understand this can someone help me
Answer:
Step-by-step explanation:
Okay, so first of all, when a slope is negative, the line goes down from left to right, not up from left to right. Then, plot the Y intercept as -2. Finally, either go up 2 and left 3 or down 2 and right 3.
Answer:
Graph attached.
Step-by-step explanation:
As we have been given the slope and the y-intercept, we can create a linear equation using the slope-intercept form.
Slope-intercept form of a linear equation:
[tex]y=mx+b[/tex]
where:
m is the slopeb is the y-interceptSubstitute the given values into the formula to create an equation for the line:
[tex]\implies y=-\dfrac{2}{3}x-2[/tex]
Now we have an equation for the line, find at least two points on the line by inputting values of x into the found equation:
[tex]x=0 \implies y=-\dfrac{2}{3}(0)-2=-2 \implies (0,-2)[/tex]
[tex]x=3 \implies y=-\dfrac{2}{3}(3)-2=-4 \implies (3,-4)[/tex]
[tex]x=-6 \implies y=-\dfrac{2}{3}(-6)-2=2 \implies (-6,2)[/tex]
Plot the found points and draw a straight line through them (see attachment).
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The slope of the line is 8. Write the point-slope equation for the line using
the coordinates of the labeled point.
The point-slope equation of the line is: y - 2 = 8(x - 2).
How to Write the Point-Slope Equation of a Line?The point-slope equation is: y - b = m(x - a), where:
m = slope(a, b) is a point.Using the coordinates of the point shown in the graph, substitute (a, b) = (2, 2) and m = 8 into y - b = m(x - a):
y - 2 = 8(x - 2)
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Graph the following inequality. y ≤ 3x +5 Use the graphing tool to graph the inequality Click to enlarge graph
Please see the graph below
Which is the equation of an asymptote of the hyperbola whose equation is [tex]\frac{(x-2)x^{2} }{4} -\frac{(y-1)x^{2} }{36}[/tex]= 1?
y = −3x − 5
y = −3x − 7
y = 3x − 5
y = 3x + 7
The equation of the asymptote is y = 3x - 5. The correct answer is option C
What is Asymptote of an Hyperbola ?The distance from a point and the distance to a line in hyperbola is known as asymptote. The general equation is [tex]x^{2}/ a^{2} - y^{2}/b^{2} = 1[/tex]
From the given equation of hyperbola, which is
[tex]\frac{(x - 2)^{2}}{4}[/tex] - [tex]\frac{(y - 1)^{2} }{36}[/tex] = 1
The center (h , k) of the hyperbola = C(2, 1)
a = 2
b = 6
Where C = [tex]\sqrt{a^{2} + b^{2} }[/tex]
C = [tex]\sqrt{4 + 36}[/tex]
C = [tex]\sqrt{40}[/tex]
C = [tex]2\sqrt{10}[/tex]
The equation of the asymptote will be y - K = +/-(b/a)(x - h)
That is,
y - 1 = +/-(6/2)(x - 2)
y - 1 = +/-3(x - 2)
y - 1 = +/-3x - 6
y = +/-3x - 6 + 1
y = +/- 3x - 5
Therefore, the equation of the asymptote is y = 3x - 5.
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If f(x)=x/4-3 and g(x)= 4x²+2x-4, find (f+g)(x).
OA. 4x²+x-7
OB.x²-12
O C. 4x²+2x+1
O D. 4x²+2x-7
Answer:
hello :
f(x)=x/4-3 and g(x)= 4x²+2x-4,
(f+g)(x).=f(x)+g(x)=x/4-3 + 4x²+2x-4,
(f+g)(x).= 4x²+(2x+x/4)-7
(f+g)(x).= 4x²+x(3+1/4) - 7......continu
Seven more than twice a number is six less than three times the same number
Answer:
13
Step-by-step explanation:
7+2x=3x-6
knowing this then you solve it
Step-by-step explanation:
7 more = + 7
twice a number = 2*x = 2x
6 less = -6
3 times the same number = 3 * x = 3x
Putting this together we get:
2x + 7 = 3x - 6
subtract two from both sides to isolate x
(-2x) + 2x + 7 = 3x - 6 (-2x)
7 = x - 6
add 6 to both sides to get your answer
7 (+6) = x - 6 (+6)
13 = x
or
x = 13
1. There are 50 contestants signed up for a TV show. There are 36 more female contestants than male contestants. How many female contestants have signed up to compete? Show your solution and explain how you plan to explain this to your students.
The number of female contestants who signed up to compete is; 43.
What is the number of female contestants?Let number of females = x
Let number of males = y.
Hence, it follows from the task content that;
x+y = 50
x-y = 36.
Hence, by adding both equations together in a bid to determine the number of females; we have;
2x = 86
x = 43.
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find the product of 3.05 and 0.07
Answer:
0.2135
Step-by-step explanation:
When something asks to find the product, it indicates multiplication.
We are given two decimals which are: 3.05 and 0.07.
We are asked to multiply and find the product of these decimals.
Let's use the algorithm method.
3 . 0 5
x 0 . 0 7
_____________________
2 3
0 . 2 1 3 5
Therefore, 3.05 × 0.07 = 0.2135.
write a piecewise function defined by three equations that has a domain of all real numbers and a range of "-3
A piece-wise function that has domain of all real numbers and range of [tex][-3, \infty)[/tex] is f(x) = |x| - 3, as:
[tex]|x| - 3 = x - 3, x \geq 0[/tex]
[tex]|x| - 3 = -x - 3, x < 0[/tex]
What is a piece-wise function?A piece-wise function is a function that has different definitions, depending on the input.
One example is the absolute value function, defined as follows:
[tex]|x| = x, x \geq 0[/tex]
[tex]|x| = -x, x < 0[/tex]
The function is defined for all real values, with a range of [tex][0, \infty)[/tex]. Shifting it down 3 units, it will have the desired range, as follows:
[tex]|x| - 3 = x - 3, x \geq 0[/tex]
[tex]|x| - 3 = -x - 3, x < 0[/tex]
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Graph line y=-1/5x-5 with y and x points
Answer:
Please see picture below.
Step-by-step explanation:
Mark the vertex and graph the axis of symmetry of the function.
f(x) = (x – 2)2 – 25
Answer:
Vertex (2 , -25)
Axis of symmetry: x = 2
Step-by-step explanation:
Vertex of parabola:The vertex is the highest point if the parabola open downwards and the lowest point if the parabola opens upward.
f(x) = (x - 2)² - 25
The given quadratic function is in vertex form.
f(x) = a(x - h)² + k
Here, (h , k) is the vertex of the parabola.
h = 2 ; k = -25
[tex]\sf \boxed{\bf Vertex (2 , -25)}[/tex]
Axis of symmetry:The axis of symmetry is the vertical line that divides the parabola into two equal halves and it passes through the vertex of the parabola.
Axis of symmetry: x = h
[tex]\sf \boxed{\bf x = 2}[/tex]
Eric's Coffee Shop makes a blend that is a mixture of two types of coffee. Type A coffee costs Eric $5.50 per pound, and type B coffee costs $4.40 per pound. This month's blend used four times as many pounds of type B coffee as type A, for a total cost of $762.30. How many pounds of type A coffee were used?
115.2 pounds of type A coffee were used.
What is the solution of the equation?A mathematical statement that has an "equal to" symbol between two expressions with equal values is called an equation.
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation. To determine the value of an unknown variable that represents an unknown quantity, equations can be solved.
Let the number of pounds of type A coffee be x and the number of pounds of Type B coffee be y.
According to the question, the equations are,
5.50x + 4.40y = 762.30
x = 4y
So, the solution of the equation is obtained as follows:
5.50(4y) + 4.40y = 762.30
26.40y = 762.30
y = 762.30/26.40
y = 28.8 pounds
x = 4*28.8 = 115.2 pounds
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How long do you need to invest your money in an account earning an annual interest rate of 4.252% compounded monthly so that your investment grows from $1,018.40 to $10,413.00 over that period of time?
The time needed for the investment to become $10,413 is 54.7 years
How long before $1,018.40 grows to $10,413.00?Number of years = (In FV / PV) / r
FV = future valuePV = present valuer = interest rater = 4.252/12 = 0.354%
(In 10, 413 / 1,018.40) / 0.00354 = 656.73 months = 54.7 years
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Convert the following equation into slope-intercept form.
7x+y = 7
y = -7x+?
Answer:
y = - 7x + 7
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
given
7x + y = 7 ( subtract 7x from both sides )
y = - 7x + 7 ← in slope- intercept form
The graph below shows the price of different numbers of mats below at a store:
Which equation can be used to determine p, the cost of b mats?
P= 10.50+b
B= 10.50+p
B= 10.50p
P=10.50b
Answer:
The answer to your question is p = 10.50b
Step-by-step explanation:
Since you want to find p, you're looking for an equation that has
p = <something>
The last selection has this form
p = 10.50b
I hope this helps and have a good day!
show a graph for the following: f(x)=3^2
The blue curve of the image attached below represents the graph of the function f(x) = 3 · x².
How to determine the graph of a given function
In this question we have a power function formed by the product of a primitive function g(x) = x² and vertical dilation h(x) = 3, then:
f(x) = h(x) · g(x)
f(x) = 3 · x²
Now we proceed to graph both the primitive function and the transformed function with a graphing tool. Please notice that the primitive function is the red curve.
Remark
The statement presents typing mistakes. Correct form is shown below:
Show a graph for the following: f(x) = 3 · x².
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The figure below shows a triangular piece of cloth: B 35 8 in. What is the length of the portion BC of the cloth? 08 cos 35° sin35° 8 cos 35° 08 sin 35°
From the calculation, we can see that /BC/ is 8 sin 35. Option D
What is the length of the portion BC of the cloth?We can see the image of the triangular piece of cloth as shown in the image. This shows that we have to approach the problem by the use of the trigonometric ratios.
Hence;
/AC/ = 8 in
/BC/ = x
<A = 35 degrees
Sin 35 = /BC//8
/BC/ = 8 sin 35
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Which of the following is not a characteristic of both observational studies and experiments
Data collected about a population is not a characteristic of both observational studies and experiments.
What is an experiment?Through experiments, two variables' cause and effect relationships are examined. This is where they differ from observations and interviews, which can only assume the existence of contexts and cannot provide evidence for them. The environment of the test subjects is managed in an experiment depending on the issue.
Three things characterize statistical experiments in general:
There are various outcomes that the experiment could produce.
In an experiment, the cause, the independent variable, is changed while the effect, the dependent variable, is measured and any unrelated factors are controlled.
It is possible to anticipate every outcome.
Chance determines how the experiment turns out.
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The height of a door is 1.5 feet longer than its width, and its front area is 1516.5 square feet. Find the width and height of the door.
Answer:
For an exact answer the width would be the square root of 1011 and the length would be 1.5x the square root of 1011.
Step-by-step explanation:
A = lw
1516.5 = w(1.5)w
1516.5 = 1.5w^2 Divide both sides by 1.5
1011 = w^2 Take the square root of each side
Square root of 1011 = w.
If the 1st and 10th terms of geometric sequence are 4 and 100, fin the nth term of the sequence.
Answer:
a = 4 first term, r = ?, T10 = 100.Tn = ar^n - 1 formula for g.pT10 = 4r^ 10 - 1100 = 4r^9Divide both side by 4100/4 = 4/4r^925 = r^9 take the 9th root of both side 9√25 = 9√r^9r = 9√25To find the nth term Since Tn = ar^n - 1
Which expression is equivalent to the following complex fraction?
1 minus StartFraction 1 Over x EndFraction divided by 2
The equivalent expression of [tex]1 - \frac{1}{x} \div 2[/tex] is [tex]\frac{2x - 1}{2x}[/tex]
How to determine the equivalent expression?The expression is given as:
1 minus StartFraction 1 Over x EndFraction divided by 2
Rewrite properly as:
[tex]1 - \frac{1}{x} \div 2[/tex]
Express the division as product
[tex]1 - \frac{1}{x} \times \frac 12[/tex]
Evaluate the product
[tex]1 - \frac{1}{2x}[/tex]
Take the LCM
[tex]\frac{2x - 1}{2x}[/tex]
Hence, the equivalent expression of [tex]1 - \frac{1}{x} \div 2[/tex] is [tex]\frac{2x - 1}{2x}[/tex]
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Michael starts a new paper company. The function fff models the company's net worth (in thousands of dollars) as a function of time (in months) after Michael starts it.
Considering the function for the net worth of the function, it is found that it will not be in debt for the first time after 6 months in month 14, at point (14,0), which is plotted on the graph.
When a company is in debt?A company is said to be in debt if it's net worth is negative. On a graph, it is represented by the curve being below the x-axis.
From month 2 until the end of month 13, the company is in debt, hence it will not be in debt for the first time after 6 months in month 14, at point (14,0), which is plotted on the graph.
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