let's recall that on the II Quadrant x/cosine is negative whilst y/sine is positive,
also let's recall that the hypotenuse is simply the radius distance and thus is never negative.
[tex]\bf cot(\theta )=\cfrac{\stackrel{adjacent}{-2}}{\stackrel{opposite}{1}}\impliedby \textit{let's find the hypotenuse} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=\sqrt{a^2+b^2} \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=\sqrt{(-2)^2+1^2}\implies c=\sqrt{5} \\\\[-0.35em] ~\dotfill\\\\ csc(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{5}}}{\stackrel{opposite}{1}}\implies csc(\theta )=\sqrt{5}[/tex]
The exact value of csc theta when the value of cot theta is -2 and the terminal side of theta lies in quadrant II (2) is √(5).
What is the terminal side of an angle?The terminal side of an angle is the rotated side of the initial side around a point to form an angle. This rotation can be clockwise or counter clock wise.
The exact value of csc theta has to be found out. The value of cot theta is,
cot θ = -2
cot θ =-2/1.
By the property of right angle triangle, the ratio of adjacent side to the opposite side is equal to the cot theta. Thus,
Adjacent side= -2
Opposite side= 1
The value of hypotenuse side is equal to the square root of the sum of the square of adjacent side and opposite side. Thus,
Hypotenuse side=√((-2)²+1²)
Hypotenuse side=√(5)
By the property of right angle triangle, the ratio of hypotenuse side to the opposite side is equal to the coses theta. Thus,
coses θ =√(5)/1
coses θ =√(5)
Hence, the exact value of csc theta when the value of cot theta is -2 and the terminal side of theta lies in quadrant II (2) is √(5).
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We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips?
There are [tex]2^{10}[/tex] possible outcomes when flipping 10 coins. Of those [tex] {10 \choose 8} [/tex] have exactly 8 heads. So the probability is
[tex]p = \dfrac{ {10 \choose 8} }{ 2^{10} } = \dfrac{10(9)/2}{2^{10}}=\dfrac{45}{1024}[/tex]
Answer: 45/1024
A) write the claim mathematically and identify upper h 0 and upper h subscript
a. (b) find the critical value(s) and identify the rejection region(s). (c) find the standardized test statistic. (d) decide whether to reject or fail to reject the null hypothesis. in a sample of 1000 home buyers, you find that 413 home buyers found their real estate agent through a friend. at alphaequals0.08, can you reject the claim that 43% of home buyers find their real estate agent through a friend?
Answer:
c
Step-by-step explanation:
Willie bought six CD's. A week later half of all his CDs were lost during a move. There are now only 20 CDs left. With how many did he start?
If Willie ends up with only 20CDs that would be half of what he had at first and 20 plus 20 which would be the other half would be 40. 6 minus 40 would be 34 which he started with.
Answer:34
Step-by-step explanation:
can someone PLEASE HELP me with this question!!!!!??
Answer:
A
Step-by-step explanation:
Repeated values in the domain of a function are not allowed.
B: -6 is repeated in the domain
C: a list of numbers, not a relation
D; 7 is repeated in the domain
__
A is a relation with no repeated domain values. It matches the requirement.
What is the value of x to the nearest degree?
25
15
65
45
[tex]\textbf{Answer}[/tex]
65 degrees
[tex]\textbf{Explanation}[/tex]
Determine which trigonomic function is required
The opposite leg is 13 and the adjacent leg is 6
We will be using tangent function
tan^-1 (13/6)
tan^-1 (13/6) = 65
for this case we have to define trigonometric relations of rectangular triangles, that the tangent of an angle is given by the leg opposite the angle on the leg adjacent to the angle. Then, according to the figure we have:
[tex]tg (x) = \frac {13} {6}\\x = arctg (\frac {13} {6})\\x = 65.22485943[/tex]
Round,
[tex]x = 65[/tex] degrees
Answer:
65
Packs of pickled peppers were reduced to 20% off their original price of $3.50. Today, there's an additional 10% off the reduced price. Find the new price for packs of pickled peppers.
Answer: $3.20
Step-by-step explanation:
$3.50 reduced by 20%= $3.30
$3.30-10%= $3.20.
A ramp with a constant incline is made to connect a driveway to a front door. At a point 4 feet from the driveway, the height of the ramp is 12 inches. At a point 6 feet from the driveway, the height of the ramp is 18 inches. What is the rate of change of the ramp's incline?
Answer:
1/4 ft vertically per 1 ft horizontally, or 3inches per foot
Step-by-step explanation:
Translate these words into symbols: One point is at (4 ft, 12 in) and the other is at (6 ft, 18 in). Now find the slope of the line connecting these two points:
As we move from (4 ft, 12 in) to (6 ft, 18 in), x increases by 2 ft and y increases by (1/2) ft.
Thus, the slope of this line is m = rise / run = (1/2 ft) / (2 ft) = (1/4) ft per ft
Answer:
D) 3 inches up per foot across
Step-by-step explanation:
Pretty sure its right
After plotting the data where t represents the number of hours since time t=0, Kieran used technology and determined the appropriate model to approximate the number of bacteria after t hours is f(t) = 5(4)t. Use the model Kieran created to predict the number of bacteria after 3 hours.
64
160
280
320
Answer:
f(t) = 5(4)^t
f(3) = 5(4^3) = 5 * 64 = 320
Jane can make a handcrafted dream catcher in 6 days. Zena makes the same dream catcher in 4 days. If they work together making dream catchers, how many days will it take to make 15 dream catchers?
Answer:
36 dream catcherss
Step-by-step explanation:
Jane's production rate:
1 dream catcher
6 days
Zena's production rate:
1 dream catcher
4 days
Production rate when working together:
1
6
+
1
4
=
5
12
. This ratio means it will take 12 days to make 5 dream catchers.
Set up a ratio for 15 dream catchers.
dream catcher
days
=
5
12
=
15
d
→ 5d = (12)(15) → 5d = 180 → d = 36
Answer: 36 days
Step-by-step explanation:
Hi, to answer this we have to calculate the production rate of each one, by dividing the number of catchers by the time spent.
Jane: 1/6 dream catcher per day
Zena: 1/4 dream Catcher per day
Adding both results:
1/6 +1/4 =5/12 dream catcher per day (both)
Finally we have to divide the 15 dream catchers needed by the rate (5/12 per day) to obtain the number of days that will take to make 15 dream catchers
x = 15 / (5/12) =36 days.
Feel free to ask for more if needed or if you did not understand something.
Give the slope and y intercept of the line determined by the equation y=2x-3
Answer:
slope= 2 y-intercept= -3
Step-by-step explanation:
the equation y=2x-3 is in slope-intercept form (y=mx+b), and m is the slope and b is the y-int. m=2 and b=-3
If f(x) = 3x + 7, which of these is the inverse of f(x)?
Answer:
f-1(x)=x-7/3
Step-by-step explanation:
The inverse of f(x) is, [tex]f^{-1}(x)=\frac{x-7}{3}[/tex]
Inverse function:Given function is, [tex]y=f(x)=3x+7[/tex]
Solve for x.
[tex]y=3x+7\\\\3x=y-7\\\\x=\frac{y-7}{3}[/tex]
Therefore, inverse function of f(x) is,
[tex]f^{-1}(x)=\frac{x-7}{3}[/tex]
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Which of the following are measures of central tendency, Thee highest valle, the lowes valle, the modelo, por the mean?
Answer:
the mode
the mean
Step-by-step explanation:
Measures of central tendency are measures of center or location of a distribution or a data set. The most commonly used measures of center of a distribution are the;
mean
median
mode.
Eric used a remainder theorem to find the remainder of 2x^3 - 4x^2 - 8 my + 1 divided by x - 3. If he calculated the remainder to be -5, what does that tell him?
The remainder is non-zero, so [tex]x-3[/tex] is not a factor of [tex]2x^3-4x^2-8[/tex] (or whatever the given polynomial is supposed to be)
The Remainder Theorem suggests that if you substitute '3' into the polynomial equation [tex]2x^3 - 4x^2 - 8x + 1[/tex], the result is -5, which is the same as the remainder of that polynomial equation divided by x - 3.
Explanation:Eric used the Remainder Theorem in his calculation, which is a mathematical principle in algebra. It states that the remainder of a polynomial f(x), when divided by a linear divisor x - a, is equal to f(a). In this scenario, when he plugged '3' (the value of 'a') into the equation [tex]2x^3 - 4x^2 - 8x + 1[/tex], he obtained a resultant value of -5.
This indicates that when the polynomial [tex]2x^3 - 4x^2 - 8x + 1[/tex] is divided by x - 3, the remainder is -5. This remainder would also be the result if 3 substituted for x in the original polynomial equation, as per the Remainder Theorem.
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A rectangular photograph measuring 2“ x 4“ is in large by a scale factor of eight what are the dimensions of the new photograph
Answer:
16 x 32
Step-by-step explanation:
If you enlarge by 8 times, you are multiplying each dimension by 8. 2*8=16 and 4*8=32
By multiplying each dimension of the original photograph (2" x 4") by the scale factor (8), we get the dimensions of the enlarged photograph as 16" (length) and 32" (width)
Explanation:The problem you presented is a simple application of scale factors in math. You need to determine the new dimensions of the enlarged photograph. For a rectangular photograph with original dimensions of 2" x 4", if the photograph is enlarged by a scale factor of eight, we multiply each dimension by the factor 8.
So, for the length: 2" x 8 = 16"
And, for the width: 4" x 8 = 32"
Therefore, the dimensions of the new photograph after enlargement would be 16" by 32".
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330 billion divided by 6.9 billion = ??
Answer:
4.782608696 x 10^10
Step-by-step explanation:
47.83.
To find the result of 330 billion divided by 6.9 billion, we can simplify the question by dividing both numbers by a billion, which gives us 330 divided by 6.9. Now, we perform the division: 330 \/ 6.9 = 47.8260869565. However, since we're dealing with significant figures usually present in population data, we might want to round the answer to a reasonable number of significant digits. So the result would be approximately 47.83.
A point has polar coordinates (r,Ф) and rectangular coordinates (x,y). If you know r and y, then:
a) it is not possible to calculate the polar or Cartesian coordinates
b) possible to get the polar coordinates but not the Cartesian coordinates
c) possible to get Cartesian coordinates, but not polar coordinates
d) possible to get Cartesian coordinates and polar coordinates
well, let's say we know "r" and also "y", can we get the angle and the rectangular "x"?
[tex]\bf \begin{cases} y=rsin(\theta )\\ x=rcos(\theta ) \end{cases} \\\\[-0.35em] ~\dotfill\\\\ y=rsin(\theta )\implies \cfrac{y}{r}=sin(\theta )\implies sin^{-1}\left( \cfrac{y}{r} \right)=\theta[/tex]
and once we know what angle θ is, since x = rcos(θ), we know "r" already, and we know θ as well, so we know "x" as well.
Knowing the radius 'r' and the y-coordinate 'y', we can calculate both, Cartesian and polar coordinates. We calculate 'x' using the Pythagorean Theorem. The angle in polar coordinates is then obtained using this 'x' and 'r' through the arccos function.
Explanation:In the context of polar coordinates and rectangular coordinates, if we know the radius 'r' and the y-coordinate 'y', we can potentially locate the specific point in the Cartesian plane. However, it's important to note that the conversion between polar and rectangular coordinates typically relies on both r and the angle Ф in polar coordinates and x, y in rectangular coordinates.
The answer to the posed question would be (d), it is possible to get Cartesian coordinates and polar coordinates.
We can get 'x' by using Pythagorean Theorem as x = sqrt(r^2 - y^2).
Similarly, the angle Ф in polar coordinates can be obtained if we know 'r' and 'x' (which we've now calculated) using the formula Ф = arccos(x / r)
Therefore, we can indeed calculate both, Cartesian and polar coordinates using the only 'r' and 'y'.
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Write the rule for finding a reflection of a point across the y-axis. Use this rule to find the coordinates for the reflection of point (?3, ?6) across the y-axis.
Answer:
The rule for finding a reflection of a point across the y-axis is that the y-coordinate will stay the same but the x-coordinate will become its opposite. Thus, in the context of the problem, point (3,6) would become (-3,6) if reflected across the y-axis.
Step-by-step explanation: If you were going to reflect a point across the x-axis, the x-coordinate would stay the same. You're not moving it and it shouldn't change signs. The y-coordinate, however, would become its opposite because it'd be joining a new quadrant. Think of it like this:
Y-AXIS = Y remains the same and gains a neighbor; X changes because it isn't the center of attention.
X-AXIS = X remains the same and gains a neighbor; Y changes because it isn't the center of attention.
Basically, the letter of the axis is the number that doesn't change.
I hope I helped!
Triangle TRI has a base of m centimeters and a height of n centimeters. Rectangle RECT has a length of n centimeters and a width of m centimeters. Use complete sentences to compare the areas of triangle TRI and rectangle RECT.
Rectangle RECT has a twice larger area than Triangle TRI. The area of Triangle TRI is mn/2 square centimeters, while the area of Rectangle RECT is mn square centimeters.
Comparing the Areas of Triangle TRI and Rectangle RECT:
Step 1: Calculate the Area of Triangle TRI:
The formula for the area of a triangle is (1/2) * base * height.
In this case, the base is m centimeters and the height is n centimeters.
Therefore, the area of Triangle TRI = (1/2) * m * n = mn/2 square centimeters.
Step 2: Calculate the Area of Rectangle RECT:
The formula for the area of a rectangle is length * width.
In this case, the length is n centimeters and the width is m centimeters.
Therefore, the area of Rectangle RECT = n * m square centimeters.
Step 3: Compare the Areas:
Now we have the areas of both shapes:
Triangle TRI: mn/2 square centimeters
Rectangle RECT: mn square centimeters
Observe that the area of Rectangle RECT is exactly twice the area of Triangle TRI.
I am being asked to calculate and plot residuals and I don't know how my x values are 13.8, 18, 16,7, 18, 0.7, 21.9, 9.2, 19.5, 15.5, 0.7 And my Y values are 1.42, 3.7, 3.21, 4, 1.11, 3.69, 2.23, 3.77, 3.92, 3.92, 1.11. And I will insert a picture of the scatterplot I made if that helps.
Answer:
(13.8,1.42) (18,3.7)(16.7,3.21) and so on REMEMBER ITS (X,Y)
Step-by-step explanation:
Which choice is equivalent to the fraction below when x is greater than or equal to 1?
Please show work.
Answer:
B. √{x} + √{x - 1}
Step-by-step explanation:
As hinted in the question, we have to simplify the denominator.
To understand it easier, let's imagine we have x - y in the denominator. If we multiply it with x + y we'll get x² - y², right? Check the next line:
(x - y) (x + y) = x² + xy -xy - y² = x² - y²
If we have the square of those nasty square roots, it will be much simpler to deal with. So, let's multiply the initial fraction using x+y, but with the real values:
[tex]\frac{1}{\sqrt{x} - \sqrt{x - 1} } * \frac{\sqrt{x} + \sqrt{x - 1}}{\sqrt{x} - \sqrt{x - 1}} = \frac{\sqrt{x} + \sqrt{x - 1}}{(\sqrt{x} )^{2} - (\sqrt{x - 1} )^{2} }[/tex]
Then we simplify:
[tex]\frac{\sqrt{x} + \sqrt{x - 1}}{(\sqrt{x} )^{2} - (\sqrt{x - 1} )^{2} } = \frac{\sqrt{x} + \sqrt{x - 1}}{(x) - (x - 1) } = \frac{\sqrt{x} + \sqrt{x - 1}}{ 1 }[/tex]
So, the answer is B. √{x} + √{x - 1}
Answer: B
Step-by-step explanation:
Which statement is true about the value (6^-n) (6^n) ?
a) for n < 0, the value of the expression is greater than 1
b) For n>0, the value of the expression 0.
c) For all n, the value of the expression is less than 1.
d) For all n, the value of the expression is 1.
Answer:d
Step-by-step explanation:
d
Follow below steps:
The student asked which statement is true about the value of the expression (6^-n)(6^n). To find the value of this expression, we can use the laws of exponents. These laws state that when we multiply powers with the same base, we add the exponents. Since one exponent is negative and the other is positive but both absolute values of the exponents are equal, their sum will be zero. So, (6^-n)(6^n) simplifies to 6^(n-n), which simplifies further to 6^0. According to the rules of exponents, any number raised to the power of zero is equal to 1. Therefore, the value of the expression is 1 for all values of n.
Two sides of a right triangle have lengths of 4 centimeters and 7 centimeters. The third side is no the hypotenuse. How long is the third side?
Answer:
√33 cm ≈ 5.745 cm
Step-by-step explanation:
Let b represent the third side. If the third side is not the hypotenuse, then the longest of the given sides must be the hypotenuse. The Pythagorean theorem tells us ...
4^2 + b^2 = 7^2
b^2 = 49 -16 = 33 . . . . . . . . subtract 16
b = √33 ≈ 5.745 . . . . . . . . . take the square root
The third side is √33 cm long, about 5.745 cm.
To find the length of the third side of a right triangle, apply the Pythagorean theorem by squaring the given sides and calculating the square root of their sum. In this case, the third side would be approximately 5.74 cm.
To find the length of the third side of a right triangle when the other two sides are 4 cm and 7 cm, you can use the Pythagorean theorem. This theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Given sides are 4 cm and 7 cm
Let's assume the third side is x
Applying the Pythagorean theorem: 4² + x² = 7²
Solving for x: 16 + x² = 49 => x² = 33 => x = √33 cm
Therefore, the length of the third side is approximately 5.74 cm
Please help out if you are any good with linear equations. Showing/explaining your work would be greatly appreciated!
Answer:
y < -2/3x +4
Step-by-step explanation:
The line intersects the y-axis at y=4, so that is the y-intercept.
The line drops 2 units for each 3 units to the right, so the slope is ...
(change in y)/(change in x) = -2/3
The slope-intercept form of the equation for a line is a form you have memorized:
y = mx + b . . . . . . for slope m and y-intercept b
Using the values we read from the graph, the equation of the line is ...
y = -2/3x + 4
The line is dashed, so the inequality does not include points on the line. The solution just includes y-values less than (below) those on the line. Hence the inequality is ...
y < (-2/3)x +4
Simplify:
5mn – 8mn + 13mn
Answer:
10mn
Step-by-step explanation:
[tex]5mn - 8mn + 13mn = 10mn[/tex]
PLEASE HELP ASAP!!! CORRECT ANSWER ONLY PLEASE!!!
Factor x^2 + 25 in the complex numbers.
Answer: C) (x + 5i)(x - 5i)
Step-by-step explanation:
x² + 25 can be rewritten as x² - (-25)
Now use the difference of squares to factor: a² - b² = (a + b)(a - b)
√x² = x √-25 = 5i
--> x² + 25 = (x + 5i)(x - 5i)
What is the common
ratio for this geometric sequence?
16,8, 4, 2, ..
Answer:
r = 1/2
Step-by-step explanation:
Each entry is diminished by a factor of 1/2 times the previous entry.
so r = 1/2
The next number in the series is 1 and then the one after that is 1/2 and then 1/4 ...
Find the sum of the arithmetic sequence. 5,7,9,11,...,23
Answer:
140
Step-by-step explanation:
The arithmetic series is 5, 7, 9, 11, ........., 23.
First u have to determine the no. of terms that can be done by using
Tₙ = [a + (n - 1)d]
Tₙ-------nth term
a---------first term
n---------no.of terms in the series
d---------common difference
here a = 5,d = 2.
let it contain n terms Tₙ= [a + (n-1)d]
Substitute Tₙ, a, and d in the equation
23 = 5 + (n - 1)2
Subtract 5 from each side.
18 = (n-1)2
Divide each side by 2
(n - 1) = 9
Add 1 to each side
n = 9 + 1 = 10
The sum of the arithmetic sequence formula: Sₙ= (n/2)[2a+(n-1)d]
Substitute Sₙ, a, n and d in the equation
Sₙ= (10/2)[2(5) + (10-1)2]
Sₙ= (5)[10 + (9)2]
Sₙ= 5[10 + 18]
Sₙ= 5[28] = 140
Therefore the sum of the arithmetic sequence is 140.
The sum of the arithmetic sequence 5, 7, 9, 11,...,23 is calculated by first finding the number of terms (10 in this case), then applying the formula for the sum of an arithmetic sequence. The sum is 140.
Explanation:The question is asking to find the sum of an arithmetic sequence consisting of the numbers between 5 and 23, increasing by 2 each time. An arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. This is the common difference. In this case, the common difference is 2.
Instead of adding each individual term, we use the formula for the sum of an arithmetic sequence. The formula is S_n = n/2 ( a + l ) where n is the number of terms, a is the first term, and l is the last term.
First, we need to find the number of terms (n). This can be found with the formula n = ( l - a ) / d + 1 where d is the common difference, a is the first term, and l is the last term. In this case, n = (23 - 5) / 2 + 1 = 10.
Therefore, the sum of this sequence is S_10 = 10/2 * (5 + 23) = 140.
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Select the conic section that represents the equation. 3x + 4y = 12 circle ellipse parabola hyperbola line
Answer:
The conic section that represents the equation 3x + 4y = 12 is a line.
3x + 4y = 12 represents the equation of the line from the given options of the conic section circle ,ellipse, parabola, hyperbola, line.
What is conic section ?" Conic section defined as the curve which is obtained when the plane is cut out from the cone."
According to the question,
Circle : Equation of the circle given by [tex](x-h)^{2} + (y-k)^{2} =r^{2}[/tex]
Conic section circle is not the correct answer.
Ellipse : Equation of the ellipse given by [tex]\frac{x^{2} }{a^{2}} + \frac{y^{2} }{b^{2}}=1[/tex]
Conic section ellipse is not the correct answer.
Parabola : Equation of the Parabola given by [tex]y^{2} = 4ax[/tex]
Conic section parabola is not the correct answer.
Hyperbola: Equation of the Hyperbola given by [tex]\frac{x^{2} }{a^{2}} - \frac{y^{2} }{b^{2}}=1[/tex]
Conic section hyperbola is not the correct answer.
Line : Equation of the line given by [tex]Ax + By + C =0[/tex]
3x + 4y = 12 represents [tex]Ax + By + C =0[/tex] .
Hence, 3x + 4y = 12 represents the equation of the line.
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How to convert from standard form to vertex form
ANSWER
See explanation
EXPLANATION
The standard form of a quadratic equation is:
[tex]y = a {x}^{2} + bx + c[/tex]
To convert this function to standard form, you follow the steps below:
Factor 'a' from the variable termsAdd and subtract the square of half the coefficient of x.Factor the perfect squaresSimplify the constant terms to get the vertex form as [tex]y = a {(x - h)}^{2} + k[/tex]For example:
Given the standard form:
[tex]y = 2 {x}^{2} + 12x + 10[/tex]
Factor 2 from the variable terms
[tex]y = 2 {(x}^{2} + 6x) + 10[/tex]
Add and subtract the square of 3.
[tex]y = 2 {(x}^{2} + 6x + 9 - 9) + 10[/tex]
[tex]y = 2 {(x}^{2} + 6x + 9) + 2( - 9) + 10[/tex]
Factor the perfect square an simplify
[tex]y = 2 ({x + 3)}^{2} - 8[/tex]
This is the vertex form
yes, one can rewrite an equation in standard form to vertex form.
Below are the steps to convert the equation in standard form to vertex form.
What do the mathematical term standard form mean?
There are various ways to represent a parabola's equation, including standard form, vertex form, and intercept form.
A parabola's typical form is standard form i.e y = ax2 + bx + c.
Here, variables x and y represent points on the parabola, and real numbers (constants) a, b, and c are real numbers (constants) where a 0.
By completing the square, change y = a (x - h)2 + k from standard form to vertex form (y = a (x - h)2 + k).
The vertex form is y = a (x - h)2 + k in this case.
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Danielle may choose one of two options for the method in which she may be awarded a money prize. OPTION A: Spin a spinner twice. The spinner is divided into four equally-sized sectors numbered 1, 4, 4, and 5. If the sum of the two spins is greater than 6, Danielle is awarded $8. Otherwise, she must pay $2. OPTION B: Flip a coin three times. If heads appears once, Danielle is awarded $6. Otherwise, she must pay $1. Danielle chooses the option with the greater mathematical expectation. How much more money can Danielle expect to make by choosing this option over the other option?
The mathematical expectation is a weighted sum:
[tex]E(X) = \displaystyle \sum_{i=1}^n x_ip(x_i)[/tex]
i.e. we multiply each outcome with its probability, and sum all these terms.
There are 16 possible outcomes for the spin, and here's table with wins/losses:
[tex]\begin{array}{c|cccc}&1&4&4&5\\1&L&L&L&L\\4&L&W&W&W\\4&L&W&W&W\\5&L&W&W&W\end{array}[/tex]
So, there are 9 winning spins and 7 losing spins. Since all the spins have the same probability, the probablity of winning $8 is 9/16, and the probability of losing $2 is 7/16. This leads to a mathematical expectation of
[tex]E(A) = 8\cdot \dfrac{9}{16}-2\dfrac{7}{16} = \dfrac{29}{8}[/tex]
In the case of the three coin flips, all triplets have the same probability of 1/8, and the eight triplets are
TTT, TTH, THT, HTT, THH, HTH, HHT, TTT
So, Danielle wins with 3 triplets, and loses with 5 triplets. The mathematical expectation is
[tex]E(B) = 6\cdot \dfrac{3}{8}-1\dfrac{5}{8} = \dfrac{13}{8}[/tex]
So, the first method is better, and the difference is 29/8-13/8 = 2.
After calculating the expected values, Danielle expects to make $0.875 more by choosing Option B over Option A, as it has a higher mathematical expectation.
To calculate which option has a greater mathematical expectation (expected value), we must consider the possible outcomes of each option and their probabilities. We then multiply each outcome by its probability and sum the results.
Option A: SpinnerWe have 3 possible sums that would result in Danielle winning $8: 5+4, 4+5, and 4+4. Any other sum results in losing $2. The probability of spinning a 5 on the first spin is 1/4, and so is the probability of spinning a 4. Therefore, the probability of getting a sum greater than 6 is:
P(sum > 6) = P(5+4) + P(4+5) + P(4+4) = (1/4)*(1/4) + (1/4)*(1/4) + (1/4)*(1/4) = 3/16.
The probability of getting a sum of 6 or less is 1 - 3/16 = 13/16. The expected value for Option A is:
E(A) = (3/16)*$8 + (13/16)*(-$2) = $0.25.
Option B: Coin FlipDanielle wins $6 if she gets exactly one head. The probability of getting exactly one head in three flips (HHT, HTH, THH) is:
P(exactly one head) = [tex](1/2)^3 + (1/2)^3 + (1/2)^3 = 3/8[/tex].
The probability of not getting exactly one head is 1 - 3/8 = 5/8. The expected value for Option B is:
E(B) = (3/8)*$6 + (5/8)*(-$1) = $1.125.
Comparing the expected values, Option B has a higher expected value than Option A. Therefore, the amount Danielle expects to make by choosing Option B over Option A is:
E(B) - E(A) = $1.125 - $0.25 = $0.875.
Danielle expects to make $0.875 more by choosing Option B.