Problem 3) A, B, and C are evenly matched tennis players. Initially A and B play a set and the winner then plays C. This continues, with the winner always playing the waiting player, until one of the players has won two sets in a row. That player is then declared the overall winner. Find the probability that A is the overall winner.

Answers

Answer 1

Answer:

the probability that A is the overall winner is 5/14 or 0.35714

Step-by-step explanation:

Given the data in the question;

probability that A is the overall winner P[A is the overall winner]

⇒ [tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{A}[/tex] [tex]P_{A}[/tex]  + ................

= [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{A}[/tex] + [tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{B}[/tex][tex]P_{C}[/tex][tex]P_{A}[/tex] + ........

P(A) = [tex]\frac{1}{2}[/tex] = [tex]P_{B}[/tex] = [tex]P_{C}[/tex]             ∴{ they are evenly matched player}

so P(A is overall winner) will be;

= ([tex]\frac{1}{2}[/tex])² + ([tex]\frac{1}{2}[/tex])⁵ + ([tex]\frac{1}{2}[/tex])⁸ + ..........+ ([tex]\frac{1}{2}[/tex])⁴ + ([tex]\frac{1}{2}[/tex])⁷ + ([tex]\frac{1}{2}[/tex])¹⁰ + .............

= ([tex]\frac{1}{2}[/tex])² [1 + ([tex]\frac{1}{2}[/tex])³ + ([tex]\frac{1}{2}[/tex])⁶ + .........] + ([tex]\frac{1}{2}[/tex])⁴[1 + ([tex]\frac{1}{2}[/tex])³ + ([tex]\frac{1}{2}[/tex])⁶ + .........]

= [ ([tex]\frac{1}{2}[/tex])² + ([tex]\frac{1}{2}[/tex])⁴ ] [ 1 / 1 - ([tex]\frac{1}{2}[/tex])³ ]

= [ ([tex]\frac{1}{4}[/tex]) + ([tex]\frac{1}{16}[/tex]) ] [ [tex]\frac{8}{7}[/tex] ]

= [tex]\frac{5}{16}[/tex] × [tex]\frac{8}{7}[/tex]

= 5/14 or 0.35714

Therefore, the probability that A is the overall winner is 5/14 or 0.35714

Answer 2

The probability that A is the overall winner is; 5/14

Let P be the chance that a player will win the tournament when he gets a new opportunity to play against the victor of the previous game.

Now, if the player loses, then it means that two games in a row are won by the same player, and then the tournament will be over. Thus; [tex]P_{L}[/tex] = 0

However, If the player wins first, and then loses his next game, it means that such a player will have to hope for a new opportunity. What this implies is that the player to whom he lost to will not win the next game. The probability of this happening is; [tex]P_{WLX}[/tex] = ¹/₂ ×  ¹/₂ ×  ¹/₂;

       [tex]P_{WLX}[/tex] = ¹/₈

Furthermore from that above, he will have another chance to win twice in a row which gives a probability in form of a geometric series as;

¹/₄ + (¹/₄ × ¹/₈ ) + (¹/₄ × ¹/₈² ) +  (¹/₄ × ¹/₈³ ) + ........

Now, for that series, as n approaches infinity, the sum of the series would give us;

∑ = ¹/₄(1/(1 - ¹/₈ ))

⇒ ¹/₄(⁸/₇)

⇒ ²/₇

Since player C's first game would already be against another game's winner, then it means that P is also the chance player C will win the tournament but the probability that player A will win is same as probability that player B will win which is;

[1 - (²/₇)]/2

⇒ ⁵/₇ × ¹/₂

P(A) = P(B) = ⁵/₁₄

Read more on probability at; https://brainly.com/question/24855677


Related Questions

3. A small construction company borrowed $500,000 to purchase some heavy machinery to prepare job sites. Some of the money was borrowed at 9% interest, some at 10% interest, and the remainder at 12% interest. The annual interest was $52,000. The amount borrowed at 10% was 2.5 times the amount borrowed at 9%. How much was borrowed at each amount?​

Answers

Answer:

they have to give money back

Step-by-step explanation:

i said so

Dennis is getting balloons for his grandmother's birthday party. He wants each balloon string to be 9 feet long. At the party store, string is sold by the yard. If Dennis wants to get 56 balloons, how many yards of string will he need?
1 foot = 1/3 yard

Answers

Answer:

168 yards

Step-by-step explanation:

3 feet = 1 yard

9 (ft for each balloon) x 56 (# of balloons.) = 504 (ft of string needed in total)

504 (ft in total) divided by 3 (three feet = 1 yard)  = 168 (yards of string needed)

Which triangle most likely has the angles 56°, 48°, and 76°?
A. triangle A
B. triangle B
C. triangle C
D. triangle D

Answers

Answer:

c

Step-by-step explanation:

At a marina there are 51 boats.
How many ways can the boats be placed in equal groups?
o
A.
2 ways
В.
4 ways
C 8 ways
D. 16 ways

Answers

Answer:

4 ways or 2 ways

Step-by-step explanation:

1 by 51

51 by 1

3 by 17

17 by 3

Please help me, thank you!​

Answers

Answer:

a = 0.3

b = 0.4

c = 40%

d = 15%

Step-by-step explanation:

There are a total of 10 tick marks. So we know that 1 tick mark = 10%

a has 3 tick marks so you can say it's 0.3

b has 4 tick marks so you can say it's 0.4

c also has 4 tick marks, but it's asking for the percentage, so you get 40%

d is 1 1/2 tick marks. It's also asking for percentage. You know 1 tick = 10% and 1/2 tick = 5* so d = 15%

24 of the cars in the parking lot are blue. about what fraction of the cars in the parking lot are blue​

Answers

Answer:

all of them are blue so 1

Step-by-step explanation:

6) How are a parallelogram and a trapezoid different?
A A trapezoid has ony 1 pair of parallel sides, but a paralelogram has 2.
C.B.A parallelogram has 4 right angles, but a trapezoid doesn't need to have 4 right
angles
CA parallelogram has only 1 pair of parallel sides, but a trapezoid has 2
D A trapezoid has 4 night angles, but a parallelogram doesn't need to have 4 night
angles

Answers

Answer:

A. A trapezoid has only 1 pair of parallel sides, but a parallelogram has 2.

That's the answer.

Answer:

a

Step-by-step explanation:

What is 2 1/3 divided by 1/2?

Also, I have the answer but is it 7/6 or 4 2/3?

Answers

the answer would be 4 2/3 or 14/3

Answer:

4 2/3

Step-by-step explanation:

hope this helps you in any way

a bag contains 3 blue and 5 red marbles find the probability of drawing 2 blue marbles in a row without replacing the first marble write as a fraction

Answers

Answer:

Step-by-step explanation:

3 Blue + 5 Red Marbles = 8 Marbles

Now 2 Blue Marbles in a row would be:

6/3

don’t know how to label the sides as “opposite, adjacent, hypotenuse” im kinda understand it but need bit support

Answers

Answer:

Opposite is on the other side to the angle θ

Adjacent is next to the angle θ

Hypotenuse is the long one.

Step-by-step explanation:

Round 6.081 to the nearest hundredth 

Answers

Answer:

6.08

Have a good day!

Convert the mixed number into a improper fraction 5 1/3

Answers

Answer:

16/3

Step-by-step explanation:

Leslie has a watering can that holds 3 liters of water. She uses 250 milliliters of water on each of her plants. How many plants can she water? Enter your answer in the box.

Answers

Answer:

12

Step-by-step explanation:

1,000 milliliters in a liter. 3 liters, so divide 3,000 by 250, which will give you 12. she can water 12 plants.

Step-by-step explanation:

Lester can water 12 plants because there is 1000 milliliters in one liter so if we do 1000x3=3000÷250=12 plants being able to get watered

Factor the expression below.
4x2 - 1

Answers

Answer:

(2x-1)(2x+1)

Step-by-step explanation:

the answer is c

The_______allowed farmers to plow fields using horses.

(heavy plow/horse collar)​

Answers

the horse collar allows the farmers to plow fields using horses

Answer:

horse collar

Step-by-step explanation:

7. Fred had 85 marbles. After a few games, he had 120% of his original number. How many
marbles does he have now?

Answers

Answer:

102 marbles

Step-by-step explanation:

100% of 85 is 85.

20% of 85 = 17

85 + 17 = 102

Hope that helps!

I need help with Geometry!

Visualize triangle ABC and triangle DEF, where AB = EF and CA = FD. What else must
be true about these two triangles to prove that the triangles are congruent
using the Hypotenuse-Leg Theorem? Write a congruence statement.

Answers

Answer:

These triangles are not fully congruent because we are missing an angle

Step-by-step explanation:

Answer:

for the triangles to be congruent using HL theorem it has to have 1 right angle

Step-by-step explanation:

ΔABC ≅ ΔDEF


anyone?............

Answers

[tex] \frac{132 }{2?} = 66[/tex]

HELp pls il give brainliest

Answers

Answer:

1. 25

2. 3

3. 10:00 am to 10:30

4. not letting you get all the answers. Try your best though the explain the answer is refering to number 3 Good luck!

Step-by-step explanation:

1) 25 miles
2) 3 hours
3) He traveled the fastest from 10:00 through 10:30 because the line show that is is steeper than any other line meaning a greater slope. Also he traveled 10 miles in 30 minutes which is a lot more than he drove for the rest of the drive left.

Write a linear equation expressing the number of t-shirts sold each month as a function of price per t-shirt. Please be explicit about what your chosen variables represent. Find the slope and intercepts of this equation. What real life meaning do these values have?

Answers

Answer:

t = c - bp

Step-by-step explanation:

Where : t = number of t shirts sold per month ,

p = price per t shirt

Equation of t shirts sold in terms of price per t shirt , ie t = f (p) , is -

t = c - bp

As per law of demand - other factors remaining constant, price & sale demand of a product are inversely related to each other.

In equation- Negative intercept of 'p' in defining 't' denotes this inverse relationship. c denotes the constant autonomous value in the equation. b denotes the slope of demand equation, which signifies demand change in response to price change.

PLEASE ASNWER ASASP FOR BRAINLEST!!!!!!!!!!!!!!!!!!!!!!!!!!

Answers

Answer:

The area is 50

Step-by-step explanation:

Trust me, I can't explain it in less than 2 book series so like :/ please rate :)

Nadine uses 6.5 pints of white paint and blue paint to paint her bedroom walls. 3 4 of this amount is white paint, and the rest is blue paint. How many pints of blue paint did she use to paint her bedroom walls?

Answers

Step-by-step explanation:if the 3 and 4 is a fraction same thing divide 6.5 by the 3 and the 4

Realiza tres ejemplos de probabilidad subjetiva

Answers

ojala te sirva

es niño o niña?: La abuela de un bebe que esta por nacer asegura que el futuro bebé será una niña y no un niño, la abuela se basa en los antojos de la madre, la forma del estómago de la madre y otras características que ella ha podido observar durante el embarazo.

te comparto imagenes ok

If n<5, then n<3 true or false pls help me?!?

Answers

I think it is false sorry if i am wrong

constant proportanalities in the equation y=rx 15,5 25,8 1/3 33,11

Answers

the constant of proportionality (r) in the equation y = rx is 1/3

PLEASE HELP AND TELL ME HOW YOU DID IT I HAVE TO SHOW WORK !!! Garys Rentals charges a 10 percent fee on all late returned vehicles. Tim returned his car late. If Tim's rental
costs $405.34 and he received a 10 percent late fee, how much did he pay in late fees?

Answers

Answer:

$40.53

Step-by-step explanation:

10% of 405.34 = 0.1 × 405.34 = 40.534

Which expression is equivalent to 20c – 8d?

2 (10c + 40)

4(50 - 8d)

4(50 – 2d)

c(20 - 8d)

Answers

hope this helped is not sorry

Can some please explain how to solve this? Thank you. 1/2(3/2y+1/3)=-1/2(y-5/2)

Answers

y = 13/15

First, write the equation:
1/2 (3/2y+1/3) = -1/2 (y-5/2)
First, distribute 1/2 by multiplying that by 3/2y and 1/3. At the same time, distribute -1/2 by multiplying that by y and -5/2. It should look like:
3/4y + 1/6 = -1/2y + 5/4
When solving to find a variable, you want to get the variable to only one side of the equation. Add 1/2y to both sides of the equation. However, when you have to add 3/4y + 1/2y, you’ll notice that the denominators are different, meaning you have to change them to make the denominators the same. Multiply 1/2 by 2 to get 2/4. Now, add 3/4 + 2/4 to get 5/4. This is what you should now have:
5/4y + 1/6 = 5/4
Next, you want to try and get the variable number by itself. As you can see, 1/6 prevents it from being by itself on one side of the equation. Subtract 1/6 from both sides. Once again, you cannot do this because the denominators aren’t the same. Multiply 5/4 by 6 and 1/6 by 4 to get 30/24 minus 4/24, which equals 26/24. You can now simplify that to 13/12. Now your equation should look like this:
5/4y = 13/12
Divide both sides of the equation by 5/4 to get y = 13/15. That is your final answer. Hope this helped! Sorry if this answer was rlly long lol

Answer:

Step-by-step explanation:

52y−13=−12 5 2 y - 1 3 = - 1 2. Combine 52 5 2 and y y . 5y2−13=−12 5 y 2 - 1 3 = - 1 2. Move all terms not containing y y to the right side of the equation.

PLS HELP IM STUCK I HAVE KARATE TO WAAAAAAAAAAAAAAAAAAAA LP WHATS THE BLANK

Answers

Answer:

Step-by-step explanation:

6 3/16

in simplified form 99/16 or 6 3/16

You are estimating the cost of optical sensors based on the resolution of the sensor. You decide to calculate the coefficient of determination (R2) as part of determining the goodness of fit of an equation. Using the preliminary calculations below, calculate the R2 and determine its meaning.
Summation of (Yi - Ybar)2 = 147172
Summation of (Yhat - Ybar)2 = 130542
Summation of (Y - Yhat)2 = 16630
a. 88.70% of the variation in the cost is being explained by the resolution.
b. 88.70% of the variation in the resolution is being explained by the cost.
c. 11.30% of the variation in the resolution is being explained by the cost.
d. 11.30% of the variation in the cost is being explained by the resolution.

Answers

Answer:

a. 88.70% of the variation in the cost is being explained by the resolution.

Step-by-step explanation:

Given :

Summation of (Yi - Ybar)2 = 147172

Summation of (Yhat - Ybar)2 = 130542

Summation of (Y - Yhat)2 = 16630

The Coefficient of determination (R²) :

Sum of Square regression (SSR) / sum of square TOTAL (SST). It gives the percentage of the dependent variable which can be explained by the regression line.

Cost is the dependent variable in this scenario.

SSR = Summation of (Yhat - Ybar)2 = 130542

SST = Summation of (Yi - Ybar)2 = 147172

Hence,

SSR / SST = 130542 / 147172

R² = 130542 / 147172

R² = 0.8870029 = (0.8870029 * 100%) = 88.7%

Other Questions
The length of a rectangle is 4 ft longer than its width.If the perimeter of the rectangle is 40 ft, find its area. What are the end behaviors of the graph f(x)=-2x+3x-4? A bottle of shampoo costs $4 at a store. It is on sale for 25% off, and the sales tax in 5.5%. What is the sales price of the shampoo? please ACTUALLY HELP>DONT JUST SPAM >IM BEGGING U>I WORKED HARD FOR THESE POINTS> Below are two parallel lines intersecting them In 728 B.C., who founded a new dynasty that ruled both Egypt and Kush? A.Kashta B.Mero C.Piye D.Axum Listen to the phone number and select the correct one: 02 20 10 11 22 02 20 01 11 22 02 20 11 10 22 02 01 20 22 11 what is the preposition in this sentenceThe children are running around the room as fast as they can. begging someone to answer this please!!! Is there anything that can be done to prevent people from falling into cheap consideration? what can we do? how can the worship of cheap material goods be stopped. Explain Lightning storms trigger wildfires that burn up a lot of the plants and grasses in the ecosystem. Trees are mostly spared. what is the cause of this?GIVING BRAINLIEST What are some of the detailed observations you will find under a cheek cell lab.Include why you think you have found certain parts of the cell. What characterized trench warfare in WWI? * What purpose is the author attempting to convey by including these quotes? She said of Trump: "He must go. He is a clear and present danger to the nation that we all love." Representative Maxine Waters, a Democrat from California, contended that Trump was "capable of starting a civil war." Which of the following returns to the beginning of the while loop?the continue statementthe break statementthe return statementthe end statement 1. Which individual(s) in the pedigree shown below must be a carrier?a. 1 onlyb. 4 only.c. 3 onlyd. both 1 and 4 why the live tutors are not available you're always not available Write a linear function f with the values f(3)=2 and f(0)=6 k^2 - 196 = 0..................................................... PLEASE HELP ME WITH THIS QUESTION Can someone help me plsssasassssss When you weigh the possible outcomes of a decision, whose well being should you take into account?1. yourself and your family2. yourself3. anyone affected by your decision4. yourself and everyone else