eight teams compete in a tournament. each pair of teams plays exactly one game against each other. there are no ties. if the two possible outcomes of each game are equally likely, what is the probability that every team loses at least one game and wins at least one game?
Answers
The probability that every team loses at least one game and wins at least one game is \(\frac{903}{1024}\).
Let’s label the 8 teams as A, B, C, D, E, F, J, H.
First, we determine the total number of games played.
Because of every pair of teams plays exactly one game, so every team plays 7 games (one against each of the other 7 teams). And there are 8 teams, so it seems as if there are 8 x 7 = 56 games, except that every game has been counted twice in this total. So, there are in fact \(\frac{8.7}{2}\)= 28 games played.
Because of there are 28 games played and there are 2 equally likely outcomes for every game, so there are \(2^{28}\) possible combinations of outcomes.
To find out the probability that every team loses at least one game and every team wins at least one game, we determine the probability that there is a team that loses 0 games or a team that wins 0 games and subtract this probability from 1.
And because we know that the total number of possible combinations of outcomes, we determine the probability by counting the number of combinations of outcomes in where there is a team that loses 0 games or a team that wins 0 games, or both.
To find out the number of combinations of outcomes in where there is a team which wins all of its games, we mark that there are 8 ways to choose this team. Once a team is chosen (we call this team X), the results of the 7 games played by X are determined (X wins all of these) and the outcomes of the remaining 28 − 7 = 21 games are undetermined.
And because of there are two possible outcomes for each of these 21 undetermined games, so there are 8 x \(2^{21}\) combinations of outcomes in which there is a team that wins all of its games. Similarly, there are 8 x \(2^{21}\) combinations of outcomes in which there is a team that loses all of its games.
Now, we note that there might be combinations of outcomes that are involved in both of these counts. There might be combinations of outcomes in where there is a team that wins all of its games and in where there is a team that loses all of its games.
Since this total has been included in both sets of 8 x \(2^{21}\) combinations of outcomes, we have to determine this total and subtract it once to leave these combinations included exactly once in our total.
To determine the number of combinations of outcomes in this case, we choose a team (X) to win all of its games and a team (Y) to lose all of its games.
Once X is selected, the outcomes of its 7 games are all found (X wins).
Once Y is selected, the outcomes of its 6 additional games are all found (Y loses these 6 games plus the game with X that has already been found).
The outcomes of the remaining 28 − 7 − 6 = 15 games are undetermined.
Therefore, the number of combinations of outcomes is 8 x 7 x \(2^{15}\) since there are 8 ways of choosing X, and then 7 ways of choosing Y (any team but X), and then \(2^{15}\) combinations of outcomes for the undetermined games.
So, there are 8 x \(2^{21}\) + 8 x \(2^{21}\) − 8 x 7 x \(2^{15}\) combinations of outcomes in where either one team loses 0 games or one team wins 0 games (or both).
Accordingly, the probability that one team loses 0 games or one team wins 0 games is
\(\frac{8.2^{21} + 8.2^{21} - 8.7.2^{15} }{2^{28} }\\\) = \(\frac{2^{15} (8.2^{6} + 8.2^{6} - 8.7)}{2^{28} }\) = \(\frac{2^{3} . 2^{6} + 2^{3} . 2^{6} - 2^{3} . 7}{2^{13}}\) = \(\frac{2^{6} + 2^{6} - 7}{2^{10}}\)
It means that the probability that every team loses at least one game and wins at least one game is 1 − \(\frac{64+64-7}{1024}\) = 1 − \(\frac{121}{1024}\) = \(\frac{903}{1024}\).
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Related Questions
Consider the following linear program.
Max 1A + 1B
s. T.
4A + 3B ≥ 24
2B ≥ 10
A, B ≥ 0
Does the linear program involve infeasibility, unbounded, and/or alternative optimal solutions? (Select all that apply. )
infeasibility
unbounded
alternative optimal solutions
Explain. (Select all that apply. )
The regions defined by the constraints do not overlap.
The first constraint allows the solution to be made infinitely large.
There are more than one solution for A and B which produce the optimal value for 1A + 1B.
The second constraint allows the solution to be made infinitely large
Answers
The options "The first constraint allows the solution to be made infinitely large" and "There are more than one solution for A and B which produce the optimal value for 1A + 1B" are correct.
The first constraint, 4A + 3B ≥ 24, defines a region in the solution space. However, this constraint does not restrict the solution to a single point or line. The feasible region can be a region of infinite possibilities, allowing for alternative optimal solutions.
The second constraint, 2B ≥ 10, also contributes to the feasibility of the problem. Since B must be greater than or equal to 0, this constraint allows for B to increase without bound.
Thus, the solution can be infinitely large along the B-axis, leading to an unbounded feasible region.
As a result, the linear program involves alternative optimal solutions due to the nature of the constraints and the possibility of an infinitely large solution along the B-axis.
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Did Sammy eat more blueberries on Monday or on
Monday or on Tuesday? Explain
your reasoning.
Answers
Answer: Sammy says he was hungry on Monday so that's why he ate more blueberries than Tuesday
Step-by-step explanation:
WILL REWARD BRAINLIEST PLS HELP ASAP Find the total surface area.
Answers
The surface area of the rectangular prism is 88 square inches.
Given that:
Length, L = 6 inches
Width, W = 2 inches
Height, H = 4 inches
Let the prism with a length of L, a width of W, and a height of H. Then the surface area of the prism is given as
SA = 2(LW + WH + HL)
SA = 2(6 x 2 + 2 x 4 + 4 x 6)
SA = 2 (12 + 8 + 24)
SA = 2 x 44
SA = 88 square inches
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Tim needs 18 pens. He can buy them in packages containing 6,9, or 12 pens each. He will buy only one type of package. Which packages could Tim buy? Write two different ways of how Tim can buy exactly 18 pens. (Goodluck)
Answers
Answer:
9 and 6
Step-by-step explanation:
You see, 9x2=18 and 6x3=18
Mia's kitten weighs 3-fifths of a pound. The mother cat weighs 14 times as many pounds as the kitten. Question How many pounds does the mother cat weigh?
Answers
So Mia kitten is 3 fifth a pound now time that 14 time: 3 x 14 = 42 so the mother cat is 42 pounds
Answer: 8.4 lbs
Step-by-step explanation:
3/5 × 14/1 = 42/5 42 ÷5= 8.4
Kiiten weighs 3/5 lbs
Mom weighs 14lbs multiple 3×14=42 then divide 42 by 5 = 8.4
What is substitute and example?
Answers
A substitute is a good or service that buyers can quickly swap out for another. For instance, a one-dollar bill can be used in place of another dollar bill.
In business and economics, a replacement, or substitutable good, is a good or service that consumers perceive as just being substantially the same as or reasonably similar to some other good. Consumers are given options and alternatives through substitutes, which also spur competition and lower prices in the market.
Here are a few examples of replacement goods:
1. A $1 bill can be exchanged for four quarters.
2. Coke against Pepsi
3. Regular vs. premium gas
4. Butter and lard
5. Tea and coffee
6. Apples and oranges
7. Comparing driving a car to riding a bike
8. Books in general and e-books
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riangle with sides of lengths 10, 24, and 27 is a right triangle.
O A. True
O B. False
Answers
Answer:
B. False
\( \sqrt{ {10}^{2} + {24}^{2} } = \sqrt{100 + 576} = \sqrt{676} = 26\)
There are 8 triangles and 4 circles. What is the simplest ratio of circles to total shapes?
Answers
The ratio of circles to total shapes is 1:3.
What is the ratio?The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.
Given:
There are 8 triangles and 4 circles.
The ratio of circles to total shapes is,
= 4/(8 + 4)
= 4/12
= 1:3
Therefore, the ratio is 1:3.
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solve for x x-1 ≥8
please help
Answers
Steps to solve:
x - 1 ≥ 8
~Add 1 to both sides
x - 1 + 1 ≥ 8 + 1
~Simplify
x ≥ 9
Best of Luck!
Square root 42 as a decimal
Answers
Answer:
6.481.
Square Root of 42 in Decimal Form: 6.481.
help !!!!!!!!!!!!!!!! plz
Answers
Answer:
5 > n + 10
8 x 7 < n
n/3 + 1
6 x 8 - 10 < n
n - 1 x n
n x -n + 7
Is 2 5/8 a rational number??
Answers
Answer:
2×5/8 which is 5/4 which is a rational number because denominator is not 0
Step-by-step explanation:
Hope it helps you
Answer:
yes it is
Step-by-step explanation:
rational numbers are numbers that can be expressed in the form of ratio. for example
\( \frac{85}{5} \)
=
85:5
To bake 12 cookies, I use 2 quarts of milk. There are 2 pints in a quart. How many pints of milk do I need to bake 3 cookies?
Answers
Answer:
1 pint
Step-by-step explanation:
You need 2 * 2 = 4 pints to make 12 cookies. Since 3 cookies is 1/4 of 12 cookies, you would need 1/4 * 4 = 1 pint of milk to make 3 cookies.
Answer:
1 pint
Step-by-step explanation:
12c=4p
divide both sides by 4
3 cookies needs 1 pint!!!!
HOPE I HELPED
PLS MARK BRAINLIEST
DESPERATELY TRYING TO LEVEL UP
✌ -ZYLYNN JADE ARDENNE
JUST A RANDOM GIRL WANTING TO HELP PEOPLE!
PEACE!
Hey you! Would you like to promote my small pastry business??
Currently we are only selling this: UR MOM (contains: Eggs, Wheat, Milk, Extra Nuts, etc” $19.99
We take cashapp, venmo, PayPal and other types of payments
Answers
Answer: nice
Step-by-step explanation:
Sounds like a Scam though ngl
you need to paint office 143. if one gallon of paint covers 50 sf, how many gallons of pant will you need?
Answers
To determine the number of gallons of paint needed to cover office 143, we need to know the square footage of the office.
Once we have that information, we can divide the square footage by the coverage rate per gallon to calculate the required amount of paint.
Let's assume the square footage of office 143 is 800 square feet.
Number of gallons needed = Square footage / Coverage rate per gallon
Number of gallons needed = 800 square feet / 50 square feet per gallon
Number of gallons needed = 16 gallons
Therefore, you would need approximately 16 gallons of paint to cover office 143, assuming each gallon covers 50 square feet.
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Solve the math question
Answers
Answer:
Step-by-step explanation:
Vertical angles:
∠1 and ∠3;
∠2 & ∠4
∠5 and ∠7;
∠6 & ∠8
Corresponding angles: angles that are on the same side of the transversal and in same position.
∠1 & 5 ; ∠2 & ∠6 ; ∠3 & ∠7 ; ∠4 & ∠8
Alternate exterior angles:
∠1 & ∠7 ;
∠2 & ∠8
Same side interior angles:
∠4 & ∠5 ;
∠3 & ∠6
Vertical Angles are :
<4,<5
<3,<6
Corresponding angles :
<1,<5
<2,<6
<4,<8
<3,<7
Alternate exterior angles:
<1,<7
<2,<8
Same side interior angles:
<4,<5
<3,<6
A shoemaker sold a pair of for $245.99 if the buyer a $300.00 bill, how much will the buyer receive in change?
*two decimal places don't forget your $ sign. Example: $50.00 NOT 50*
Answers
Answer:
$54.01
Step-by-step explanation:
All you have to do is $300.00-$245.99 .
Aki made 190 tote bags. she sold 115 of the at the craft fair for a total of $2,070.00. which equation can be used to find the price ,p, for which she wilde a tote bag?
a. 190x 115 = p
b. 190p = 115
c. 190p=$2,070.00
d. 115p = $2,070.00
e. 75p= $2,070.00
Answers
The equation 115p = $2,070.00 can be used to find the price ,p, for which she buy a total bag. So the option d is correct.
Aki made 190 total bags.
She sold 115 of the at the craft fair for a total of $2,070.00.
To solve a word problem, you must first read the problem and identify the key facts. Once you have identified the key facts, you can use those facts to set up an equation or a system of equations that can be used to solve the problem. Finally, use the equations or system of equations to solve the problem.
Let the cost of one bag is x.
So the cost of 115 bags is 115x.
From the question total cost of the 115 bags is $2070.00.
So the equation should be
115x = $2070.00
So the option d is correct.
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What kind of function is
5x + 19y = 84 ? Thanks
Linear, slope-Intercept form Quadratic
exponential
linear, standard form
Answers
Answer:
It's a linear function.
3 John can clean pools at a constant rate of What is the ratio of pools to hours? pools per hour.
Answers
-x^2 = -36
Solve the equation using inverse operations. Check your solutions. In you final answer, include all of your work.
Answers
Answer:
x = ±6
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDASEquality PropertiesAlgebra II
Extraneous solutions and multiple answers/rootsStep-by-step explanation:
Step 1: Define
-x² = -36
Step 2: Solve for x
Divide -1 on both sides: x² = 36Square root both sides: x = ±6Step 3: Check
Plug in x to verify it's a solution.
Substitute in -6: -(-6)² = -36Exponents: -(36) = -36Multiply: -36 = -36Substitute in 6: -(6)² = -36Exponents: -(36) = -36Multiply: -36 = -36Here we see that both -6 and 6 do indeed work as solutions.
∴ x = ±6 are both solutions to the equation.
can ya find the area of this triangle
Answers
\(\sf \:Formula \: to \: find \: the \: area \: of \: a\: triangle \: = \boxed{\boxed{\bf \frac{1}{2} \times b \times h}}\)
___________________
\(\sf \: b = 6 \: in. \\ \sf \: h = 6 \: in.\)
___________________
\(\sf \: Area \: \hookleftarrow \\ \sf \: = \frac{1}{2} \times b \times h \\ \sf \: = \frac{1}{2} \times 6 \times 6 \\ = \sf \frac{1}{\bcancel2} \times{ \bcancel36 } \\ \sf =\underline{\bf 18 \: in ^{2} .}\)
An ANOVA procedure is applied to data obtained from four distinct populations. The samples, each comprised of 15 observations, were taken from the four populations. The degrees of freedom for the numerator and denominator for the critical value of F are ________.
Answers
The degrees of freedom for the numerator and denominator for the critical value of F are _3 and 56, respectively_.
The samples, each comprised of 15 observations, were taken from the four populations.
How to find the degrees of freedom?The determine the degrees of freedom for the numerator and denominator for the critical value of F is given below:
k = 4
n = 15
Total degree of freedom;
= nk - 1
= 59
For numerator, it is
= k -1
= 4 - 1
= 3
For denominator it is
= T - (k -1 )
= 59 - 3
= 56
The degrees of freedom for the numerator and denominator for the critical value of F are _3 and 56, respectively_.
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Jessica’s car depreciates in value by 11% per year. Jessica bought her car for £21000. Work out how much her car will be worth after 7 years, giving your answer to the nearest penny.
Answers
Answer:
Brainliest
Step-by-step explanation:
21,000 x 0.89^7 =
9,288.58032806109 = 9288.58
Answer:
£9288.58
Step-by-step explanation:
Initial price= 21000
Price drop rate= 11%
Time= 7 years
Price at the end of 7 years: £21000*((100-11)/100)^7= £9288.58
Solve 3x - 5 = 16
Write your solution starting with x= .
Answers
Answer:
x = 7
Step-by-step explanation:
3x - 5 = 16
3x = 16 + 5
3x = 21
x = 21/3
x = 7
I
Eugenia quiere comprar un departamento de interés social
que cuesta $320,520.00. En la inmobiliaria, le ofrecen: la
mitad al contado y el resto en 12 cuotas fijas iguales. ¿Cuál
es el valor de la cuota?
Answers
Answer:
The answer is "13355 pesos per installment"
Step-by-step explanation:
The social representation division spends $320,520, But it offers half money, which means we'll pay for half at $320,520, which is equal to $160,260. It divided into equal-sized fixed installment payments but we said, that we took half at $32,520, which would be equal to $13,355 pesos, which would be equal to $1,355 per installment.
Whats 1+1+2+3+7+4+9=?
Answers
Answer:
27
thanks for the points
Question 18
What is the quotient of (x² - 4x - 32) ÷ (x − 8) ?
Answer
(F)-8x + 4x
G+4
H-4x + 4
Jx-4
Answers
Answer:i need more info
Step-by-step explanation:
what is the slope of the following graph?
Answers
Answer: 4
Step-by-step explanation:
from point (0,-4), it goes up 4 and over 1, ending at point (1,0)
Answer:
4
Step-by-step explanation:
count the amount of spaces(or lines) to where the second point is(4) and the go right (1) 4/1 simplified is 4.
plz mark brainiest =)
which of these equivalent to the expression 5y÷7×-2y÷3×
Answers
Answer:
\(D) 3y+10x\)
Step-by-step explanation:
→ \(5y+7x-2y+3x\)
→ \(5y-2y+7x+3x\)
→ \(3y+10x\)
\(----------\)
\(hope~ it ~ helps...\)
\(have ~ a ~ great ~ day!!\)