The expressions that represents Carissa's distance to the right of the center of the skitrail, i.e, h in terms of the number of times Carissa has traveled since she started skiing,d is h = 1.3 cos( d/1.3). So, correct answer is option (c).
We have, Carissa skis along a circular ski trail. Radius of circle , r = 1.3 km
Carissa starts at the 3-o'clock position of the trail travels in the counter clockwise direction. let "d" km be total distance that Carissa has traveled since she started skiing and "h" km be Carissa's distance to the right of the center of the skitrail in km.
We have to wright an expressions for "h" in terms of d . Here, we see an sector is formed with one side length is r and arc's distance is d km . Let θ be angle formed by this sector . Using the sector formula,
θ = arc length /radius of circle
=> θ = d/r = d/1.3 --(1)
Also, ∆ formed here , h = r cos θ
=> h = 1.3 cos( d/1.3)
Hence, the required expression is h=1.3 cos( d/1.3).
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Complete question:
Carissa skis along a circular ski trail with a radius 1.3 km long. Carissa starts at the 3-o'clock position of the trail (the East-side) and travels in the CCW direction. Which of the following expressions represents Carissa's distance to the right of the center of the skitrail in km, h, in terms of the number of km that Carissa has traveled since she started skiing.d?
a)h 1.3 cos(d+ 1.3)
b) h = 1.3 cos(1.3/d)
c)h = 1.3 cos (d /1.3 )
d)h = 1.3 cos(d)
e) h = COS( 1.3 )
Submit Question 16. Points possible:
Question 6 of 21
What are the vertex and x-intercepts of the graph of the function given below?
y=x²-2x-24
A. Vertex: (-1, -21); intercepts: x = 6,4
B. Vertex: (0, 0); intercepts: x=-4, 6
C. Vertex: (7, 5); intercepts: x = 6,8
OD. Vertex: (1, -25); intercepts: x= 6,-4
Answer:
OD=Vertex: (1, -25); intercepts: x= 6,-4
Step-by-step explanation:
y=x²-2x-24
where a = 1,b = –2 and c = –24
to find the vertex we will use this formula
Step 1
(h,k) = (-b/2a, -D/4a), where D = b²–4ac.
(h,k) = (–(–2)/2×1) , –((–2)² – 4 × 1 × –24/4×1)
(h,k)= (2/2) , –(4 + 96)
(h,k) = (1 , –100/4)
(h,k) = (1,–25)
STEP 2
To find x intercept make y = 0
y=x²-2x-24
x² – 2x – 24 = 0
x² –6x + 4x – 24 = 0
x(x –6) +4(x –6) = 0
(x+4)(x–6)=0
x+4 = 0; x = –4
x–6 = 0; x = 6
x = (–4,6)
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For the following exercises, express each function H as a composition of two functions F and G where H(x) = (fog)(x)
42) The functions F and G are G(x) = (2x - 1)/ (3x + 4) and F(x) = √x
43) The functions F and G are G(x) = 3x² - 4 and F(x) = 1/x⁻³.
What is a function?
A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The sets X and Y are collectively referred to as the function's domain and codomain, respectively.
Given composite function is
H(x) = (fog)(x).
42)
H(x) =√[(2x - 1)/ (3x + 4)]
Rewrite the above equation
H(x) =√[G(x)] where G(x) = (2x - 1)/ (3x + 4)
H(x) = F(G(x)) where F(x) = √x
42)
H(x) =1/(3x² - 4)⁻³
Rewrite the above function
H(x) =1/[G(x)]⁻³ where G(x) = 1/(3x² - 4)
H(x) = F(G(x)) where F(x) = 1/x⁻³
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write an expression for the sequence of operations described below. triple s, subtract u from the result, then subtract t from what you have do not simplify any part of the expression.
The expression for the sequence of operations is 3s - u - t .
In the question ,
it is given that ,
the sequence of operation is given as " triple s, subtract u from the result, then subtract t from what you have" .
we have to write an expression for the above operation ,
So ,
first step is : triple s ,
that means the expression is 3s .
next subtract u from the result , that means ⇒ 3s - u .
next we have to subtract t from what we have ,
that means ⇒ 3s - u - t .
Therefore , the final expression is 3s - u - t .
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two people are selected at random from a group of 16 republicans and 19 democrats. find the probability of each of the following. (round your answers to three decimal places.)
The probability of both people being Democrats is 0.243, and the probability of one person being a Republican and one person being a Democrat is 0.486.
To find the probability of each of the following, we can use the formula for the probability of selecting a group of items from a larger group with replacement:
P(A and B) = P(A) * P(B)
Where P(A) is the probability of selecting the first item, and P(B) is the probability of selecting the second item.
Both people are Democrats:
There are 16 Democrats and 31 total people, so the probability of selecting a Democrat is 16/31. Therefore, the probability of selecting two Democrats is (16/31) * (16/31) = 0.243.
One person is a Republican and one person is a Democrat:
There are 15 Republicans and 31 total people, so the probability of selecting a Republican is 15/31. There are 16 Democrats and 31 total people, so the probability of selecting a Democrat is 16/31. Therefore, the probability of selecting one Republican and one Democrat is (15/31) * (16/31) = 0.486.
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Let X and Y be IID Unif(0, 1). Find the expected value and the standard deviation of X-Y. i.e., the distance between X and Y.
Let X and Y be IID Unif(0, 1). the distance between X and Y.Expected value: 0 and Standard deviation: [tex]$\sqrt{\frac{1}{3}}$[/tex]
Expected value:
E(X-Y)
= E(X) - E(Y)
= 0 - 0
= 0
Standard deviation:
Var(X-Y) = Var(X) + Var(Y) = 1 + 1 = 2
SD(X-Y) = [tex]$\sqrt{2}$[/tex]
= [tex]$\sqrt{\frac{1}{3}}$[/tex]
The expected value of X-Y is the difference of the expected values of X and Y, which are both 0 since they are both IID Unif(0, 1). The variance of X-Y is the sum of the variances of X and Y, which are both 1. Since the standard deviation is the square root of the variance, the standard deviation of X-Y is the square root of 2.
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g a florida health insurance company wants to predict annual claims for individual clients. the company pulls a random sample of 50 customers
The Florida health insurance company can use a variety of predictive methods to predict the annual claims for the sample of 50 customers. These methods include machine learning algorithms such as linear regression and decision trees, as well as statistical methods such as regression analysis and logistic regression.
The company can also use qualitative methods such as interviews and surveys to gain insight into customer needs and preferences. By combining quantitative and qualitative methods, the company can create a more accurate prediction of the annual claims for its sample of 50 customers.The company can calculate its predictions by first gathering data on each of the 50 customers, such as age, gender, health history, lifestyle, and other factors. It can then use this data to create a predictive model that can be used to estimate the annual claims of each customer. These methods include machine learning algorithms such as linear regression and decision trees, as well as statistical methods such as regression analysis and logistic regression. The model can be tested and refined as necessary to ensure it is accurate and reliable. Finally, the company can use the model to make its predictions for the 50 customers.
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What is the answer?
For the given circle the value of cotθ is -√30/√19.
What is Trigonometry?Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.
The terminal side of an angle θ.
The standard position intersects the unit circle at (√30/7, -√19/7).
We need to find the value of cotθ.
Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at (√30/7, -√19/7)
cotθ=x/y
=-√30/7/√19/7
=-√30/7×7/√19
=-√30/√19
Hence, the value of cotθ is -√30/√19 for the given circle.
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Write the equation of the line, in all three forms, that is parallel to the line that passes through the points (-6, 1) and (10,0) and passes through point (-7,-2).
The linear equation parallel to the line that passes through (-6, 1) and (10,0) and that passes through (-7, -2) is:
y = (-1/16)*x - 39/16
How to find the linear equation?
A general linear equation is written as:
y = a*x + b
Where a is the slope and b is the y-intercept
If a linear equation crosses through two known points (x₁, y₁) and (x₂, y₂), then the slope of that line is given by:
a = (y₂ - y₁)/(x₂ - x₁)
And two lines are parallel if and only the two lines have the same slope and different y-intercept.
Here we want to find a line that passes through (-6, 1) and (10, 0), the slope of that line is:
a = (0 - 1)/(10 + 6) = -1/16
And we want to find a line with this slope that passes through the point (-7, -2), so we need to have:
-2 = (-1/16)*-7 + b
-2 = 7/16 + b
-2 - 7/16 = b
-32/16 - 7/16 = b
-39/16 = b
So the linear equation is:
y = (-1/16)*x - 39/16
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on my knees begging pls im tired and exhausted
A) If 3/5 of the 595 students are seventh graders, and 5/8 of them participated in the competition, about how many seventh graders participated in the competition? Describe the process you used to find your answer. (2 points)
b) If 2/5 of the 595 students are eighth graders, and 7/8 of them participated in the competition, about how many eighth graders participated in the competition? Describe the process you used to find your answer. (2 points).
c) About how many total students participated? Describe the process you used to find your answer. (2 points) Using the process shown above, I added 206 and 215 to get 421 total participants
a) The number of seventh graders that participated in the competition was of 223.
b) The number of eight graders that participated in the competition was of 208.
c) The total number of students that participated in the competition is of 431.
How to obtain the amounts?The amounts in this problem are obtained applying the proportions given by the fractions.
For item a, we have that the amount is 5/8 of 3/5 of 595, hence the total amount is obtained as follows:
5/8 x 3/5 x 595 = 223 students.(rounding to the nearest integer).
For eight graders in item b, we have that 7/8 of 2/5 of 595 students participated, hence the number of students that participated in the competition is given as follows:
7/8 x 2/5 x 595 = 208 students. (rounding to the nearest integer).
Then the total number of students that participated in the competition is given by the sum of these amounts as follows:
223 + 208 = 431 students.
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7x-8y=10 and x-8y=-26 using elimination
x= 6;
y= 4.
Step-by-step explanation:1. Write the equations.[tex](1)7x-8y=10\\\\ (2)x-8y=-26\\[/tex]
2. Subtract equation 2 from equation 1.[tex](1)7x-8y=10\\-\\ (2)x-8y=-26\\--------\\\\6x+0y=36\\ \\6x=36[/tex]
3. Solve for x.[tex]\frac{6x}{6} =\frac{36}{6} \\ \\x=6[/tex]
4. Use the calculated value of "x" in any of the equation to the value of "y".For this solution, we are going to use equation 2 to get the value of "y".
[tex](6)-8y=-26\\\\-8y=-26-6\\ \\-8y=-32\\ \\\frac{-8y}{-8} =\frac{-32}{-8} \\ \\y=4[/tex]
5. Verify.If the answers are correct, both equations should return the same value in both sides of the equal (=) sign.
[tex]7(6)-8(4)=10\\ \\42-32=10\\ \\10=10[/tex]
That's correct.
[tex](6)-8(4)=-26\\ \\6-32=-26\\ \\-26=-26[/tex]
That's correct.
Hence, the answers are correct.
In 1989, the average age of a divorced man was 36.2 years of age, according to data obtained from the U.S. Department of Health and Human Services. A sociologist wants to investigate if the average age of a divorced man is now different. She randomly selects 32 divorced men and finds that the mean age is 32.5 years with a sample standard deviation s = 9.6. At a 95% level of confidence, we wish to investigate if the average age of a divorced man is now different than it was in 1989.
24. State the Null and Alternate Hypothesis for this study.
A) H0: µ = 33.9 H1: µ 33.9
B) H0: µ = 36.2 H1: µ 36.2
C) H0: µ >= 33.9 H1: µ <33.9
D) H0: µ >= 36.2 H1: µ < 36.2
On solving the provided question, P = 0.037 reject H o,P-value < 0.05 ( level of significance )
What is a null hypothesis?If there is no statistical significance between two variables, the null hypothesis is that there isn't any. The researcher or experimenter generally aims to refute or undermine the theory. An alternate explanation is that the two variables have a statistically significant association.
based on P value -
P = 0.037 reject H o
P-value < 0.05 ( level of signifance )
we are 95% y confidence that the average age of divorced man is how different than it was in 1989.
we would have concluded that the average age is not different wher, it really was different.
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A consumer group claims that the mean annual consumption of high fructose corn syrup by a person in the United States is 48.8 pounds. A random sample of 120 people in the United States has a mean annual high fructose com syrup consumption of 49.5 pounds. Assume the population standard deviation is 3.6 pounds. At alpha = 0.05, can you reject the claim? Identify the null and alternative hypotheses. Identify which hypothesis is the claim. Use the sample statistics given to find the standardized test statistics: z and P. Make a decision to reject or fail to reject the null hypothesis. Interpret your decision in the context of the original claim.
The null hypothesis in the given question is 'the mean annual consumption of high fructose corn syrup by a person in the United States is 48.8 pounds' and the alternative hypothesis in the given question is 'the population standard deviation is 3.6 pounds.'
Population mean = 48.8 pounds
Sample size = 120
Sample mean = 49.5 pounds
Population standard deviation = 3.6
α = 0.05
The standardized statistic z is:
z = (Sample mean - Population mean) x (√n / standard deviation)
z = (49.5 - 48.8) x (√120/3.6)
z = 2.13
The standardized statistic P is:
P = 2 - 2 x Φ(z)
P = 2 - 2 x 0.983
P = 0.034
Since, the result we get is statistically significant. We reject the null hypothesis.
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Allam just finished a great meal at a restaurant in Wisconsin. The sales tax in Wisconsin is 5%, and it is customary to leave a tip of 15% for the server. The tip amount is calculated on the price of the meal before the tax is applied. (Sales tax is not calculated on tips.)
If Allam ordered a $15 meal, how much money will he have to pay with tax and tip?
Answer:
$18
Step-by-step explanation:
15 : 100 = x : 15
x = 15 · 15 / 100
x = 225 / 100
x = $ 2.25 (tips)
5 : 100 = x : 15
x = 15 · 5 / 100
x = 75 / 100
x = $ 0.75 (tax)
15 + 0.75 + 2.25 = $18
f(x)=x²-8x³ +26x² - 88x - 111; x = −11
Answer:
14772
Step-by-step explanation:
3. Net income = gross profits - (cost to bring goods into store + ___)
A. long-term liabilities
B. total assets
C. operating expenses
D. cost of merchandise sold
Answer: D & C
Step-by-step explanation:
The net Income of the business is represented as
Net profits or income = Net sales revenue (gross income) – Cost of goods and services sold – Administration, operating, marketing, and advertising costs, taxes, interests, and other expenses.
Here C is an option because we need to take operating expenses into consideration but if you consider cost of goods brought into store as operating then option changes to D.
D represents the sales which helps us determine the net income of the business.
Right Answers Only! thank you
Answer: A
Step-by-step explanation:
D is wrong because its saying 3 times of the flat fee of 52.5 per month.
B is wrong because it is saying 52.5$ per gigabyte
C is wrong because the total amount is not 64.2
The polynomial function p(x)=x³-3 x²+x+205 has x=-5 as a root.
What are the remaining roots of p(x) ? Separate multiple answers with a comma.
Enter the factored form of p(x).
The remaining roots of p(x) is x = -5 , 4 + 5i , 4 - 5i and the factored form of p(x) = ( x + 5 ) ( x - ( 4 + 5i ) ( x - ( 4 - 5i )
Given :
The polynomial function p(x)=x³-3 x²+x+205 has x=-5 as a root.
using synthetic division :
x = -5
5 1 -3 1 205
0 5 10 55
1 2 11 260
on solving using the quadratic formula or calculator the roots are :
x = -5 , 4 + 5i , 4 - 5i
so quadratic form of p ( x ) = ( x + 5 ) ( x - ( 4 + 5i ) ( x - ( 4 - 5i )
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A rectangular living room has a perimeter of 40 meters. It’s area is 100 square meters.what are the dimensions of the living room
Answer:
Step-by-step explanation:
Firstly, I presume that the question talks about a rectangle and there is consistency in units i.e.Area is in units^2 and perimeter is in units (else there is no possible solution).
Now, perimeter = 2(L+B) = 40
OR (L+B) = 20.
Also, area = LxB = 100.
Now, LxB = 100 will give you the following values for LxB (Note in a rectangle, L>B or L=B):- (100×1),(50×20),(25x4),(20×5),(10×10).
Only one value gives L+B = 20 viz.L=10 & B=10.
Hope this helps! :)
(This was by another user you can search it up and see.)
How many miles is it from Austin Texas to Seattle Washington by car?
Answer:
The total driving distance from Austin, TX to Seattle, WA is 2,128 miles
Step-by-step explanation:
Answer:
i believe it is a 2,116.4 mile drive.
Step-by-step explanation:
compared to smaller firms that export, larger companies tend to multiple choice systematically scan foreign markets to see where export opportunities lie. delay exporting until after their domestic market is fully saturated. be intimidated by the complexities and mechanics of exporting to foreign countries. be reactive about seeking opportunities for profitable exporting. hesitate to seek export opportunities because they do not know how big the opportunities actually are.
According to the profit point of view, compared to smaller firms that export larger companies tend to systematically scan foreign markets to see where export opportunities lie.
What does profit mean?
In math, profit means the excess of returns over expenditure in a transaction or series of transactions.
Here we have given that, compared to smaller firms that export, larger companies tend to to be what.
And for the reason more firms are not proactive is that they are unfamiliar with foreign market opportunities; they simply do not know how big the opportunities actually are or where they might lie. Simple ignorance of the potential opportunities is a huge barrier to exporting
Here we know that, smaller firms, are often intimidated by the complexities and mechanics of exporting to countries where business practices, language, culture, legal systems, and currency are very different from the home market
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4) How long will Lourdes have to wait for her plant to grow a total 14 1/4 inches
Answer:
10 [tex]\frac{5}{28}[/tex] days
Step-by-step explanation:
There are a lot of places to make mistakes, but I think that I am good.
Make a proportion:
[tex]\frac{Days}{Inches}[/tex] = [tex]\frac{Days}{Inches}[/tex] Put in the three number that you know and solve for the number of days.
[tex]\frac{7\frac{13}{21} }{10\frac{2}{3} }[/tex] = [tex]\frac{x}{14\frac{1}{4} }[/tex] Cross multiply and solve for x
10 [tex]\frac{2}{3}[/tex] x = [tex]7\frac{13}{21}[/tex]([tex]14\frac{1}{4}[/tex]) Change everything into improper fractions
[tex]\frac{32}{3}[/tex]x = ([tex]\frac{160}{21}[/tex])[tex]\frac{57}{4}[/tex] I am going to cross cancel the 160 and 4 They both can be divided by 4 and I am going to cross cancel the 21 and 57. They are both divisible by 3
[tex]\frac{32}{3}[/tex] x =( [tex]\frac{40}{7}[/tex])[tex]\frac{19}{1}[/tex]
[tex]\frac{32}{3}[/tex] x = [tex]\frac{760}{7}[/tex] Multiply both sides by [tex]\frac{3}{32}[/tex]
([tex]\frac{3}{32}[/tex])[tex]\frac{32}{3}[/tex]x =([tex]\frac{3}{32}[/tex])[tex]\frac{760}{7}[/tex] I can cross cancel 32 and 760. They both can be divided by 8
x = ([tex]\frac{3}{4}[/tex])[tex]\frac{95}{7}[/tex]
x = [tex]\frac{285}{28}[/tex]
x = 10 [tex]\frac{5}{28}[/tex] Days
PLEASE ANSWER ASAP
WILL MAKE BRAINLIEST
A 30 kg object takes one MINUTE to travel east 240 m. It began travelling at 240 m/s and came to a stop (0 m/s) at the end. What was its acceleration?
Responses
1
-4
8
-0.125
Answer:
-4 im pretty sure
Step-by-step explanation:
= -4
= -4
15. The formula for the area A of a triangle is A = bh, where b is the base
and h is the height. Solve the formula for h and justify each step. Then
find the height of a standard yield sign when the area is 558 square inches
and each side is 36 inches.
taking . definitions: a nullcline of this system is a line on which either or . more precisely, a vertical nullcline of this system is a line on which , and likewise, a horizontal nullcline of this system is a line on which . (a) write an equation for a vertical nullcline that is not a coordinate axis:
The a) equation for the vertical nullcline is y = -4x + 4 and horizontal nullcline is x = -3y + 1. The b) equilibrium points are (0,0) , (0 , 0.8) , (0.75 , 0), and (0.375,0.5)
We have
dx/dt = x(1 - x/3 - y)
dy/dt = y(1 - y/4 - x)
a) To get the equation for vertical nullcline, not on the coordinate axis we take
dy/dt = 0
or, y(1 - y/4 - x) = 0
or, 1 - y/4 - x = 0
or, y/4 = 1 - x
or, y = -4x + 4
Similar;y for horizontal nullcline not on coordinate axis we take
dx/dt = 0
or, (1 - x/3 - y) = 0
or, x/3 = 1 - y
or, x = -3y + 1
b)
To get the equilibrium points of the nullcline system we simply take
Here the x nullcline will intersect y nullcline at
x = 0 and x = 1 - x/3 - y
y = 0 and y = 1 - y/4 - x
therefore we have equilibrium points
for x = 0 and y = 0 at(0,0)
x = 0 and y = 1 - y/4 - x at (0, 4/5)
x = 1 - x/3 - y and y = 0 at (3/4 , 0)
x = 1 - x/3 - y and y = 1 - y/4 - x at (3/8 , 1/2)
Hence the equilibrium points are (0,0) , (0 , 0.8) , (0.75 , 0), and (0.375,0.5)
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Complete Question
(Image Attached)
Suppose your waiting time for a bus in the morning is uniformly distributed on [0, 12], whereas waiting time in the evening is uniformly distributed on [0, 14] independent of morning waiting time.(a) If you take the bus each morning and evening for a week, what is your total expected waiting time? (Assume a week includes only Monday through Friday.) [Hint: Define rv'sX1, , X10and use a rule of expected value.]min(b) What is the variance of your total waiting time? (Round your answer to two decimal places.)min2(c) What are the expected value and variance of the difference between morning and evening waiting times on a given day? (Use morning time − evening time. Round the variance to two decimal places.)expected value minvariance min2(d) What are the expected value and variance of the difference between total morning waiting time and total evening waiting time for a particular week? (Use morning time − evening time. Assume a week includes only Monday through Friday.)expected value minvariance min2
The total expected waiting time including the morning and the evening time is 41/3
given that your waiting time for a bus in the morning is uniformly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time.
Sum of both waiting times = X+Y
Where X = morning wait time is U(0.8) and
Y = evening wait time is U(0,10)
Since X and Y are independent
Var(x+y) = Var(x)+Var(y)
Var(x) = (8^2 - 0^2)/12 = 16/3
Var(Y) = (10^2 - 0^2)/12 = 25/3
Var(x+y) = 16/3 + 25/3
= (16 + 25)/3
= 41/3
Therefore, the total expected value of waiting time including the morning and the evening time is 41/3.
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Raúl is pouring a concrete walkway in his backyard. He decides to put a turn of 30° from the horizontal in the walkway to go around a tree and flowerbed. Once the walkway is past the tree, Raúl wants to continue the walkway in the same direction as before. At what angle should he put the second turn so that the walkway will continue in the same direction as before it reached the flowerbed? Why?
Answer:
Step-by-step explanation: Raúl should put a turn of -30° in the walkway after the tree in order to continue in the same direction as before. This is because the negative angle will cancel out the positive angle and the walkway will continue in the same direction as before.
Question 2
What is the length of DE? (number only)
The length of the side DE, that is parallel to the side AC is 6.3 units
Here the line DE is parallel to the side AC of the triangle
In the given triangle
The length of the side BD = The length of the side AD
The length of the side BE = The length of the side EC
That mean the parallel line AC is equally dividing the two side of the triangle
Therefore, the length of the line DE will be be exactly half of the line AC
The length of the side AC = 12.6 units
The length of the side DE = The length of the side AC / 2
Substitute the values in the equation
= 12.6 / 2
= 6.3 unit
Therefore, the length of DE is 6.3 units
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Question
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
We still need to find a way to add these fractions. Now you try.
Change the denominators to rename these fractions and find their sum.
1/2+1/3=?
After changing the denominators to rename these fractions the sum is 5/6 .
Given :
If necessary, use / for the fraction bar(s).
We still need to find a way to add these fractions. Now you try.
Change the denominators to rename these fractions and find their sum.
To find sum we need to find LCM of denominators .
Least common multiple :
Least Common Multiple ( LCM ) is a method to find the smallest common multiple between any two or more numbers.
so = 1 / 2 + 1 / 3
LCM of 2 and 3 is 6
= 1 * 3 + 1 * 2 / 6
= 3 + 1 * 2 / 6
= 3 + 2 / 6
= 5 / 6
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i need to know how to do this
Alice and Bob are playing a game where in each round, each of them rolls a fair die independently. If both roll the same number, then the game is repeated. Otherwise the player with the larger number wins. Let X be the number of rounds until the game is decided.
(a) Determine the probability mass function of X.
(b) Compute E[X].
(c) Compute Probability[ Alice win].
(d) Assume that you get paid 10USD for winning in the first round, 1USD for winning in any other round, and nothing otherwise. Compute your expected winnings.
(a) The probability mass function of X will be P(X=n) = 5/(6n)
(b) E[X] =1.2
(c) Probability [ Alice win] will be =0.5
(d) The value of expected winnings will be 4.25
(a) Let 'p' represent the probability of getting the same number on both the dices on a roll of pair of die
And (1-p) = probability of not getting the same number on both the dices on a roll of pair of die
Sample space for the same number appearing on both the dices on a roll of pair of die = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}
As there are a total of 36 outcomes on a roll of pair of dice,
p = 6/36 = 1/6
Let the events be defined as:
Dn : Both Alice and Bob roll the same number on the nth roll
Wn : The game is finished in the nth roll (i.e. Alice and Bob roll different numbers (as the numbers are different, one will be larger))
Then,P(W) = P(W) , n = 1,2,3,4....infinity
Wn takes place when event D takes place in the first (n-1) rolls and Event W takes place in the nth roll
[tex]$\mathrm{P}(\mathrm{W})=\sum P\left(W_n\right), \mathrm{n}=1,2,3,4 \ldots .$[/tex]. infinity
And, W_n takes place when
Event D takes place in the first (n-1) rolls and event W takes place in the [tex]$n^{\text {th }}$[/tex] roll
[tex]P\left(W_n\right)=P\left(D_1\right)^* P\left(D_2\right)^* P\left(D_3\right) \ldots . .{ }^* P\left(D_{n-1}\right) P\left(W_n\right)P\left(D_n\right)\\=(p)P\left(W_n\right)\\=(1-p)[/tex]
Implies
[tex]$$\begin{aligned}& \left.P\left(W_n\right)=(p)^{n-1}\right)^*(1-p) \\& =\left((1 / 6)^{n-1}\right)^*(1-(1 / 6)) \\& =\left((1 / 6)^{n-1}\right)^*(5 / 6) \\& =5 /\left(6^n\right)\end{aligned}$$[/tex]
(b)
[tex]$$\begin{aligned}& \mathrm{E}[\mathrm{X}]=\sum n * P(X=n), \mathrm{n}=1,2,3,4 \ldots . . \text { infinity } \\& =\sum n * \frac{5}{6^n} \\& =5 * \sum n * \frac{1}{6^n}\end{aligned}$$[/tex]
If [tex]$\mathrm{S}=\sum n * \frac{1}{6^n}$[/tex]
Then S is a sum of infinite aritho-geometric series of the form
[tex]$$\sum_{k=1}^{\mathrm{inf}} k * r^k \text {, }$$[/tex]
The sum of which is given as [tex]$S=\frac{r}{(1-r)^2}$[/tex] for 0 < r < 1
Here, r=(1 / 6)
[tex]\Rightarrow & S=\frac{\frac{1}{6}}{\left(\frac{5}{6}\right)^2} \\& =\frac{6}{25}[/tex]
E[X] = (5) x (6/25) = (6/5)= 1.2
(c) Since the dies are fair, and the outcomes are independent, the probability of each player winning is equal
Therefore,
P(Alice WIn) = P(Bob WIn) = 0.5
(d) If X represent the round in which the game ends, then the probability of us winning in that round
P(W|X=n) = 0.5*P(X=n)
= (0.5)*(5/(6n))
P(W|X=1) = (0.5)*(5/(61)) = (5/12)
P(W|X=2,3,4...) = 0.5 - P(W|X=1) = (1/2) - (5/12) = (1/12)
Leth the variable W represent the winnings
(W|X=1) = 10
(W|X=2,3,4...) = 1
E[W] = P(W|X=1)*(W|X=1) + P(W|X=2,3,4...)*(W|X=2,3,4...)
= (10 x (5/12)) + (1 x (1/12))
= (50/12) + (1/12)= (51/12)= 4.25
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