Sample Spaces and Probabilities (18 points) For each of the following scenarios first describe the sample space and indicate how big it is (i.e., what its cardinality is) and then answer the question. (a) [3 Points] You flip a fair coin 50 times. What is the probability of exactly 20 heads? (b) [3 Points] You roll 2 fair 6-sided dice, one red and one blue. What is the probability that the sum of the two values showing is 4? (c) [3 Points] You are given a random 5 card poker hand (selected from a single deck). What is the probability you have a full-house (3 cards of one rank and 2 cards of another rank)? (d) [3 Points] 20 labeled balls are placed into 10 labeled bins (with each placement equally likely). What is the probability that bin 1 contains exactly 3 balls? (e) [3 Points] There are 30 psychiatrists and 24 psychologists attending a certain conference. Three of these 54 people are randomly chosen to take part in a panel discussion. What is the probability that at least one psychologist is chosen? What is the probability that exactly three psychologists are chosen? (f) [3 Points] You buy ten cupcakes choosing from 3 different types (chocolate, vanilla and caramel). Cupcakes of the same type are indistinguishable. What is the probability that you have at least one of each type?

To show that if the column vectors of matrix A and B are linearly independent, then the column vectors of matrix C = A*B will also be linearly independent, we can use the following proof:

Assume that the column vectors of matrix A and B are linearly independent. Let's assume that the column vectors of matrix C = A*B are linearly dependent.

This means that there exists a non-trivial linear combination of the column vectors of C that results in the zero vector.

C = AB => c_1 = Ab_1, c_2 = Ab_2, ... c_n = Ab_n

Let's assume that a linear combination of c1, c2, ..., cn results in the zero vector.

a_1c_1 + a_2c_2 + ... + a_n*c_n = 0

Now, we can substitute the matrix multiplication for ci:

a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n = 0

Since matrix A is non-singular, we can left multiply by A^-1

A^-1*(a_1Ab_1 + a_2Ab_2 + ... + a_nAb_n) = A^-1*0

This simplifies to

a_1b_1 + a_2b_2 + ... + a_n*b_n = 0

This is a non-trivial linear combination of the column vectors of matrix B that results in the zero vector, which contradicts the assumption that the column vectors of matrix B are linearly independent. Therefore, the column vectors of matrix C = A*B must also be linearly independent.

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Using the greatest common factor the expression is factored to be

The largest number among all the common factors of the given numbers is known as the GCF (Greatest Common Factor) of two or more numbers.

The greatest integer that may be used to divide two natural numbers x and y without producing any remainders is called their GCF.

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.

The greatest common factor of 48 and 72 is 12

hence we have

12 (4 + 6)

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Answer:

224 total cookies sold

Step-by-step explanation:

if 9 sugar with every 5 cholates then, 144/9 = 16

so there were 16 times 5 cookies were sold with means

80 cholate cookies were sold

80+ 144= 224

:)

Answer:

1015

Step-by-step explanation:

29 es un número primo.

Por tanto:

29*35 = 1015

El mínimo común múltiplo de 29 y 35 es:

1015

Sheila get $77 and Dora get $121 paid.

The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.

Given:

Sheila and Dora worked 3 1/2 hours and 5 1/2 hours.

Total they get paid = $198.

In one hour the amount of work they done is 7/2 and 11/2.

So, the ratio of their work = (7/2)/ (11/2)= 7/11

So, the amount Sheila get

= 7/18 x 198

= $77

and, the amount Dora get

= 11/18 x 198

= $121

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Answer:

2/11

Step-by-step explanation:

Normally 22 kids. 22 kids on the attendance roster.

18 kids say, "Here!" when their name is called.

22 - 18 is 4. 4 kids don't say anything when their name is called because they are at home still. 4 kids absent.

4 out of 22 kids absent. 4/22 kids absent.

4/22 simplifies to 2/11.

2/11 of the kids are absent.

I have 40 cupcakes in my house. That day, 3 friends came over. All of us have the same number of cupcakes each. How many cupcakes did we each eat?

The story of 40 divided by 4 reminding everyone about the importance of fairness and sharing.

There were 40 colorful candies waiting to be shared equally among 4 friends.

The candies were so delicious and enticing that the friends couldn't wait to indulge in their sweet treat.

The four friends, named Alex, Ben, Chris, and Dana, gathered around a table with their empty candy bags in front of them.

They were filled with excitement as they knew it was time to divide the candies fairly.

With the candies in the center of the table, they decided to take turns dividing them into smaller piles.

Alex went first and carefully divided the candies into 10 piles, making sure each pile had an equal number of candies.

This left 10 candies in each pile.

Then, it was Ben's turn. He took one pile of 10 candies and divided it into two equal piles, with 5 candies in each.

Ben did the same with the other nine piles, resulting in a total of 20 piles, each containing 5 candies.

Next, it was Chris's turn. He picked one pile with 5 candies and divided it into two equal piles, leaving 2 candies in each pile.

Chris repeated this process with the remaining 19 piles, resulting in a total of 40 piles, each containing 2 candies.

Finally, it was Dana's turn. Dana took one pile with 2 candies and divided it equally between two empty candy bags, leaving 1 candy in each bag. Dana repeated this process with the remaining 39 piles, resulting in a total of 80 candy bags, each containing 1 candy.

In the end, all four friends had received their fair share of candies. They joyfully took their candy bags and started savoring the delicious treats, grateful for their successful division.

And so, the story of 40 divided by 4 came to a sweet conclusion, reminding everyone about the importance of fairness and sharing.

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Answer:

One bar each

Step-by-step explanation:

8 people/8granola bars =1 bar each

The difference between the area of triangle ABC and the area of triangle EFJ is 0 square unis

From the question, we have the following parameters that can be used in our computation:

Trapezoid ABCD and trapezoid EFGH are congruent.

From the complete question added as an attachment

We can see that the two triangles have the same base and the same height due to the congruence of the two trapezoids

This means that their areas are equal

The difference between equal values is 0

Hence, the area difference is 0

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The reasons that matches each steps are:

1. Given (first occurrence)

2. Given (second occurrence)

3. Vertical angles theorem

4. ASA

5. CPCTC

CPCTC stands for "corresponding parts of congruent triangles are congruent", which connote that if two triangles are congruent, then every parts that are corresponding will be congruent to each other also.

Thus the reason that match each steps would be stated as follows:

1. C is the intersection point of CD and EB » Given (first occurrence)

2. AC ≅ EC and <A ≅ <E » Given (second occurrence)

3. <ACB ≅ <ECD » Vertical angles theorem

4. ∆ABC ≅ ∆EDC » ASA

5. AB ≅ ED » CPCTC

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This is an example of descriptive statistics. Descriptive statistics is used to summarize, organize, and present data in a meaningful way.

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It is proved that there are exactly [tex]2^{\frac{n(n-1)}{n}}[/tex] labeled simple graphs on n vertices. And about [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex]  labeled simple graphs that have exactly m edges.

(i) The number of edges that can be on n vertices is [tex]\frac{n(n-1)}{2}[/tex]. We have (n-1) choices to put an edge for each vertex. Also, we have n choices to select the first vertices. So in total, we have n(n-1) choices but each edge is counted twice. Then, all distinct edges on n vertices are given by  [tex]\frac{n(n-1)}{2}[/tex].

Let us consider H to be any graph that lies on n vertices. And this denotes all the edges [tex]e_1, e_2, e_3,\cdots \cdots \cdots e_{\frac{n(n-1)}{2}}[/tex]. For each edge [tex]e_i[/tex], the number of choices we have is two. One is [tex]e_i[/tex] is an edge in H. Another is [tex]e_i[/tex] is not an edge in H. This is therefore written as 2×2×..............×2 choices, and this number [tex]2^{\frac{n(n-1)}{n}}[/tex] equals labeled simple graphs on n vertices.

(ii) Out of [tex]\frac{n(n-1)}{2}[/tex] edges, we can select m edges for a graph by [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex]ways. Then, the number of graphs that are present on n vertices with m edges is written as [tex]\left(\begin{array}{c}{ \frac{n(n-1)}{2}}&\\m&\end{array}\right)[/tex].

The complete question is -

(i) Show that there are exactly [tex]2^{\frac{n(n-1)}{n}}[/tex] labeled simple graphs on n vertices.

(ii) For any value of, how many of those labeled simple graphs have exactly m edges?

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The length of AC of the given right angle triangle is; AC = 8

Pythagoras Theorem is defined as the way in which you can find the missing length of a right angled triangle. The triangle has three sides, the hypotenuse, Opposite and the adjacent side.

Now, the given triangle is a right angle triangle and using Pythagoras theorem, we can say that;

AC = √(BC² - AB²)

Thus;

AC = √(10² - 6²)

AC = √64

AC = 8

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If Ben scores 56 out of 79 on a maths test. The  percentage he get is: 0.7089.

Using this formula to determine the percentage he got

Percentage gotten = Number of scores / Total score

Where:

Number of score = 56

Total score = 79

Let plug in the formula

Percentage gotten = 56 ÷ 79

Percentage gotten = 0.70886

Percentage gotten = 0.7089 (Approximately)

Therefore we can conclude that he got 0.7089 .

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The value of x which makes the equation true is 15.

What is the system of equations?

A system of equations is a set of two or more equations that contain multiple variables. These equations are typically used to describe a real-world situation and are solved simultaneously to find the values of the variables.

To solve a system of equations for unknown variables, you can use one of several methods, such as substitution, elimination, or graphing.

To solve a system of equations such as x-4=11, you have to find the value of x which will make the equation true.

x-4=11

x=15

Hence, the value of x which makes the equation true is 15.

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Random sampling is the sampling method that was used.

Random sampling is a method in which all members of the population have an equal chance of being selected for the sample. In this case, fifteen instructors were chosen using random numbers to implement the new learning strategy.Random sampling is a useful tool when a researcher needs to select a representative sample from a population. By using random numbers to choose the sample, the researcher can ensure that the sample is truly representative of the population. Random sampling also helps to reduce bias, as the researcher has no control over who is chosen for the sample.

To calculate the probability of being chosen for the sample, the formula is (1/n) x 100, where n is the total number of population members. In this case, the population was the group of all instructors, and the sample was the group of fifteen instructors chosen. Therefore, the probability of being chosen is (1/n) x 100 = (1/15) x 100 = 6.67%.

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Answer:

The gutters are 24 ft above the ground. For safety, the distance a ladder reaches up a wall should be four times the distance from the bottom of the ladder to the base of the side of the house. Therefore, the ladder must be 6 ft from the base of the house.

The domain of a function refers to the set of all possible inputs for the function.

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Sentence A: "The joke was so hilarious that the audience roared!" most accurately and vividly describes the action.

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Sentence A is the most accurate and vivid as it describes the audience's reaction in a lively and emotional way.

In this sentence, the verb "roared" is used to describe the reaction of the audience to the joke. Roaring is a strong and lively verb that conveys the idea that the audience was highly entertained and had a strong emotional reaction to the joke.

Sentence B: "The joke was so hilarious that the audience smirked!" is not vivid enough to describe the action as the audience reaction is not clearly defined. Smirking is usually a subtle expression that might be difficult to interpret.

Sentence C: "The joke was so hilarious that the audience smiled!" is also not vivid enough to describe the action as smiling is a common reaction to something funny and it doesn't clearly convey the level of audience's reaction.

Overall, Sentence A is the most accurate and vivid as it describes the audience's reaction in a lively and emotional way.

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The probability that Federer missed his first serves given that he won the point is 0.30%.

Generally, To solve this problem, we can use Bayes' theorem. Bayes' theorem states that:

P(A|B) = P(B|A) * P(A) / P(B)

where

A and B are events, and

P(A|B) is the probability of event A given that event B has occurred.

In this problem, let A be the event that Federer missed his first serve, and let B be the event that he won the point.

We are trying to find P(A|B), the probability that Federer missed his first serve given that he won the point.

We know that:

P(A) = 1 - 0.63 = 0.37 (the probability that Federer missed his first serve, given that he made 63% of his first serves)

P(B|A) = 0.57 (the probability that Federer won the point given that he missed his first serve)

P(B|A') = 0.78 (the probability that Federer won the point given that he made his first serve)

We can use these probabilities and Bayes' theorem to find the answer:

P(A|B) = P(B|A) * P(A) / ( P(B|A) * P(A) + P(B|A') * P(A'))

P(A|B) = 0.57 * 0.37 / ( 0.57 * 0.37 + 0.78 * 0.63)

= 0.30029 or 30.33%

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Answer:

0.3

Step-by-step explanation:

Tennis great Roger Federer made 63% of his first serves in a recent season. When Federer made his first serve, he won 78% of the points. When Federer missed his first serve and had to serve again, he won only 57% of the points. Suppose you randomly choose a point on which Federer served. You get distracted before seeing his first serve but look up in time to see Federer win the point. What’s the probability that he missed his first serve?

The fraction of water left from the bottle which was 1 liter is 3/4 liter.

Fraction simply means a piece of a whole. In this situation, the number is represented as a quotient such that the numerator and denominator are split. In this situation, in a simple fraction, the numerator as well as the denominator are both integers.

In this case, from the bottle which was 1 liter, Juan drank 1/4. The amount that is left will be:

= 1 - 1/4

= 3/4

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The measure of  the angles <ABC and <CBD are 48⁰ and 42⁰

The angle ABC and angle CBD are complementary angles, m <ABC=(3x+6)⁰ and m<CBD=(4x-14)⁰

The angles are complementary which means that they sum to 90⁰

Therefore, we get: <ABC + <CBD = 90⁰

3x + 6 + 4x - 14 = 90 ------ Substituting 3x + 6 and 4x - 14 for < ABC and <CBD

7x = 98

x= 98/7 = 14⁰

Therefore; Measure of <ABC:

Measure of<CBD:  

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Correct question:

Find the measure of each angle angle ABC and angle CBD are complementary angles, m angle ABC=(3x+6)degrees, and m angle CBD=(4x-14)degrees stuck on this problem, Thank you fo

(a) (i) Average velocity = -160 ft/s, (ii) Average velocity = -320 ft/s, (iii) Average velocity = -1600 ft/s. (b) Instantaneous velocity = -320 ft/s.

(a) (i) Average velocity = change in height/change in time

= (285 - (285 - 16(2.1)^2))/0.1

= -160 ft/s

(ii) Average velocity

= (285 - (285 - 16(2.05)^2))/0.05

= -320 ft/s

(iii) Average velocity

= (285 - (285 - 16(2.01)^2))/0.01

= -1600 ft/s.

(b) Instantaneous velocity = -320 ft/s.

The average velocity of the pebble after t seconds is given by the equation y = 285 − 16t2. To find the average velocity for the time period beginning when t = 2, we can calculate the change in height of the pebble over the given time period and divide it by the change in time. For instance, for 0.1 seconds, we can calculate the average velocity by dividing the change in height (285 - (285 - 16(2.1)^2)) by the change in time (0.1). This gives us an average velocity of -160 ft/s. Similarly, for 0.05 and 0.01 seconds, the average velocity can be calculated to be -320 ft/s and -1600 ft/s respectively. To estimate the instantaneous velocity of the pebble after 2 seconds, we can assume that the change in time is very small and approximate the velocity to be the average velocity for 0.01 seconds. This gives us an instantaneous velocity of -320 ft/s.

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a) The inequality equation is x < -11

b) The inequality equation is x < 7

c) The ratio of dogs to cats is 7 : 5

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the inequality equation be represented as A

Now , the value of A is

a)

Substituting the values in the equation , we get

x + 8 < -3   be equation (1)

On simplifying the equation , we get

Subtracting 8 on both sides of the equation , we get

x < -11

Therefore , the inequality is x < -11

b)

The inequality representing the number line is x < 7

c)

Let the ratio be represented as A

The number of dogs = 7

The number of cats = 5

The ratio of dogs to cats = number of dogs / number of cats

Substituting the values in the equation , we get

The ratio of dogs to cats = 7 / 5

Therefore , the ratio is 7 : 5

Hence , the inequality relation is solved

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The criterion that she should use to prove that triangles RST and XYZ are congruent are:

i. Hypotenuse Leg (HL)

ii. Side-Angle-Side (SAS)

When the corresponding properties i.e. length of sides and measure of internal angles of two or more triangles are equal, then the triangles are said to be congruent. The relations can be shown using some theorems.

From the given triangles RST and XYX, it can be observed that;

TR ≅ XZ

RS ≅ XY

<R ≅ <X   (given)

<T ≅ <Z   (given)

Angles R and X are included angle of the triangles. Thus, the criterion that can be used to show that triangles RST and XYZ are congruent at;

i. Hypotenuse Leg (HL)

ii. Side-Angle-Side (SAS)

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Slope is a metric that expresses how inclined a line is with respect to the horizontal.Answer is y = (-5/3)x - 1 .

Slope is a metric that expresses how inclined a line is with respect to the horizontal.

The anti-clockwise measured tangent of the angle between a line and the positive x-axis is the slope of a straight line.

The slope of the line will be m = tan if is the angle between a straight line and the positive x-axis.

Y/X = (Y2 - Y1)/(X2 - X1) = m = tan

A straight line's equation is (y - y1)/(x - y1) = (y2 - y1)/(x2 - x1) if it goes through the points (x1, y1) and (x2, y2).

Y = mx + c is the equation of a straight line in slope intercept form.

Solution:

Given: A straight line connects (-3, 4) and (0, -1).

To determine: The straight line equation

Since ,

Since (-3,4) and (0,-1) are on a straight line, its equation will be as follows:

y - y1 / x - x1 = y2 - y1 / x2 - x1

→ (y - 4)/(x - (-3)) = (-1 - 4)/(0 - (-3))

→ (y - 4)/(x + 3) = - 5/3

→ y - 4 = (-5/3)(x + 3)

→ y - 4 = (-5/3)x - 5

→ y = (-5/3)x - 5 + 4

→ y = (-5/3)x - 1

Hence ,

The necessary formula is y = (-5/3)x - 1.

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Answer:

The two numbers are 17 and 7

Step-by-step explanation:

Translate this word problem into a math equation

If we let x and y be the two unknown numbers

One-half the sum of two numbers is 12 ==>

So one equation is 1/2(x + y) =12

Multiplying throughout by 2 gets rid of the denominator:
2 x 1/2 x (x + y) = 2 x 12

x + y = 24   (1)

One of the numbers is three more than two times the other number:
x = 2y + 3

or

x - 2y = 3  (2)

Equation (2) - Equation (1) =>

x + y - (x -2y) = 24 - 1
x + y - x + 2y = 21

3y = 21

y = 21/3

y = 7

With y = 7, substituting in equation 1 gives

x + y = 24

x + 7 = 24

x = 24-7

x = 17

The two numbers are 17 and 7

Answer: To find the two integers between which the solution to 4^x = 50 lies, we can take the logarithm base 4 of both sides of the equation: x = log4(50). The logarithm of 50 with base 4 is approximately 2.3219. Since the logarithm of a number is always positive, the solution to 4^x = 50 lies between 2 and 3.

To find the exact solution, we can change the equation to logarithmic form by taking the logarithm base 4 of both sides: log4(4^x) = log4(50)

Using the logarithmic identity: log(b^x) = x*log(b)

So we have: x*log(4) = log(50)

Dividing both sides by log(4), we have:

x = log(50) / log(4)

To find the exact solution we can approximate log(50) / log(4) using a calculator or logarithm tables. The exact solution is x ≈ 2.32192809488736218170856773213.

So, the exact solution is x = 2.32192809488736218170856773213, which is between 2 and 3, as stated before.

Step-by-step explanation: