Gerolamo Cardano in his book, The Gambling Scholar, written in the early
1500s, considers the following carnival game. There are six dice. Each of the
dice has ve blank sides. The sixth side has a number between 1 and 6|a
dierent number on each die. The six dice are rolled and the player wins a
prize depending on the total of the numbers which turn up.
(a) Find, as Cardano did, the expected total without nding its distribution.
(b) Large prizes were given for large totals with a modest fee to play the
game. Explain why this could be done.

answer: Mean:5

median:4

range: 9

mode: 2

Step-by-step explanation:

Answer:

74.8 m²

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7.6 cm}\underline{Area of a triangle} \\\\$A=\dfrac{1}{2}ab \sin C$\\\\where:\\ \phantom{ww}$\bullet$ $C$ is the angle. \\ \phantom{ww}$\bullet$ $a$ and $b$ are the sides enclosing the angle. \\\end{minipage}}[/tex]

From inspection of the given triangle:

Substitute the values into the formula and solve for area:

[tex]\implies A=\dfrac{1}{2} \cdot 11 \cdot 15 \cdot \sin 115^{\circ}[/tex]

[tex]\implies A=\dfrac{165}{2} \sin 115^{\circ}[/tex]

[tex]\implies A=74.77039243[/tex]

[tex]\implies A=74.8\; \sf m^2\;(nearest\;tenth)[/tex]

Therefore, the area of the triangle is 74.8 m² rounded to the nearest tenth.

The value of the missing angle of the triangle is; x = 31°

Alternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent.

Now, from the diagram, we can tell that ∠GAB is congruent to ∠CBA by alternate angles theorem.

Thus, ∠GAB = 28°

Similarly, sum of angles on a straight line = 180°. Thus;

∠EDA = 180 - (4x + x)

∠EDA = 180 - 5x

Also, sum of angles in a triangle is 180 degrees. Thus;

∠CAB = 180 - (90 + 28)

∠CAB = 62°

Thus;

62 + 3x + 180 - 5x = 180

62 - 2x = 0

2x = 62

x = 62/2

x = 31°

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Opposite sides of a parallelogram are equal.

Opposite sides of a parallelogram are equal in the second case

RT is congruent to RT is obvious.

ΔRST and ΔRUT are congruent because a diagonal of a parallelogram divides it into two congruent triangles.

We know pair of interior alternate angles are equal.

We know two similar planer figures are congruent when we have sides or angles or both that are the same as the corresponding sides or angles or both.

Given, Line segment TU is congruent to line segment RS because in a parallelogram opposite are equal same reason for line segment RU and ST.

We also know that the diagonal of a parallelogram divides the parallelogram into two equal triangles hence ΔRST and ΔRUT are congruent.

We also know that the pair of alternating angles are equal hence,

∠RST and ∠TRU are congruent.

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The linear equation obtained is y = 0.3X + 18.3

A linear equation is an equation that can be written in the form ax + b = 0, where a and b are real numbers and x represents an unknown variable. The solutions of linear equations are often graphed on a coordinate plane, forming a straight line. Linear equations can be used to model many real-world situations, such as population growth, the rate of change of a physical quantity, or the total cost of an item. Linear equations can also be used to solve systems of equations, where several equations must be solved simultaneously.

Given that data :
x = year ;
y = trunk diameter, in inches
Year ________trunk diameter
1 __ 18.6
3 __ 19.2

5 __ 19.8
7 __ 20.4
9 ___21.0
11 __ 21.6
13 __ 22.2
Using the linear regression calculator :
The linear equation obtained is :
y = 0.3X + 18.3
Where ;
Slope = 0.3 ; intercept, c = 18.3

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Answer:

y - 8 = 2 (x - 6)

Step-by-step explanation:

The point-slope equation is ...

y - y1  = m ( x - x1 )

In the coordinate (6,8)

x = 6 and y = 8

m = 2

Plug in values to find the point-slope form

y - 8 = 2 (x - 6)

We are done!

The expression 2 3/5 ÷ 5 3/8 given is written in simplest form as  104/215

The problem is a division problem in mixed numbers

converting to improper fractions we have

= 2 3/5 ÷ 5 3/8

= 13/5 ÷ 43/8

inverting the divisor to change to multiplication

= 13/5 ÷ 43/8

= 13/5 * 8/43

= 104/215

The division problem is simplest form give 104/215

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The differential equation that can represent all circles in the xy plane is given as follows:

[tex]\frac{dy}{dx} = -\frac{x - h}{\sqrt{r^2 - (x - h)^2}}[/tex]

The equation of a circle of center (h,k) and radius r is given as follows:

(x - h)² + (y - k)² = r².

The equation can be arranged to isolate the variable y as follows:

(y - k)² = r² - (x - h)²

y - k = square root (r² - (x - h)²)

y = square root (r² - (x - h)²) + k

To obtain the differential equation, then this equation must be differentiated towards variable x as follows:

[tex]\frac{dy}{dx} = \frac{d}{dx}(\sqrt{r^2 - (x - h)^2} + k)[/tex]

Applying the chain rule, along with the derivative for the square root, we have that:

[tex]\frac{d}{dx}(\sqrt{r^2 - (x - h)^2} + k) = -\frac{x - h}{\sqrt{r^2 - (x - h)^2}}[/tex]

Meaning that the differential equation is given as follows:

[tex]\frac{dy}{dx} = -\frac{x - h}{\sqrt{r^2 - (x - h)^2}}[/tex]

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Answer: x = 1, y = -1

Step-by-step explanation:

      To solve this system by elimination, we will set it up so both of the y, or x, variables have opposite coefficients.

3x - 5y = 8

2x+ 3y = -1

3(3x - 5y = 8)

5(2x+ 3y = -1)

9x - 15y = 24

10x + 15y = -5

      The 15y and -15y cancel each other out, so we're left with 9x + 10x on the left side and 24 - 5 on the right side of the equation.

19x = 19

x = 19/19

x = 1

      Now, we will plug this back into one of the equations to solve for y.

3x - 5y = 8

3(1) - 5y = 8

3 - 5y = 8

-5y = 5

y = -1

The randomized variable's sample distribution would be as follows: H+/- 1.96 * √3)/√n

A random variable in mathematics would be a variable for whom the value is the numerical result of an unpredictable event and a random number generator X on a sample space. S denotes the rule that gives each of the outcomes S in a collection a numerical value.

In this case, we have assumed that x is a constant random variable with values ranging from 0 to 3, but we are unsure of its distribution, mean, or variance.

The mean of x, which we designate by the letter h, is what we're now interested in estimating. To provide a conservative estimate (upper bound), we'll choose 1.5 for the standard deviation of x.

In this case, we must estimate h. To do this, we select the sample size n, the samples x1, x2,.., xn, all of which share the same dissemination as x, and accurately measure the sample mean.

Therefore, we have used a normal distribution to determine the variation of x as (0,6),

And it equals to 3, then apply CLT resulting in

=> H+/- 1.96 * √3)/√n

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Correct option is C

In order to understand the relation between smoking and hypertension (HT) a tea of researchers designed a study that looks at the numbers of smokers and nonsmokers with no HT, pre-HT, stage 1 HT, and stage 2 HT.

An independent samples t test should be run for the study because the outcome smoker vs nonsmoker is a categorical variable and the grouping variable no HT, pre-HT, stage 1 HT, and stage 2 HT-categorizes the sample into four independent groups.

Variables

In Maths, a variable is an alphabet or term that represents an unknown number or unknown value or unknown quantity. The variables are specially used in the case of algebraic expression or algebra. For example, x+9=4 is a linear equation where x is a variable, where 9 and 4 are constants.

Probability

Probability is simply likely something is to happen. Whenever unsure about the outcome of an event, we can talk about the probabilities of certain outcome. The analysis of events governed by probability is called statistics.

Correct option is C

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Answer:1/4 (I think)

By using Dijkstra's algorithm, it can be concluded that-

It will loop forever.

Third option is correct

What is Dijkstra's algorithm?

Dijkstra's algorithm is used to find the shortest path between the nodes of a graph.

Here, a node is fixed which is called the source node and the distance from the source node to all other nodes are calculated.

The minimum distance gives the shortest path.

For the first option,

The first option is incorrect because the shortest path will not be obtained. There does not exist path between every pair of vertices

For the second option

The second option is incorrect because all the conditions of Dijkstra's algorithm are fulfilled but here, infinite looping takes place.

For the fourth option

The fourth option is incorrect because there exist a path between the point M and every other vertex in the graph.

So the third option is correct

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Complete Question

The complete question has been attached

From the the graph f(x) = -x-3

Part a

The value of f(-4) = 1

Part b

The value of f(5) = -8

Part c

The value of f(0) = -3

The function is given that

f(x) = -x-3

The function is the mathematical statement that shows the relationship between the independent variable and dependent variable. The function consist of the different variables, numbers and mathematical operator.

The function is

f(x) = -x - 3

Part a

The value of f(-4)

Substitute the value of x in the function

f(-4) = - (-4) - 3

= 4 - 3

= 1

Part b

The value of f(5)

f(5) = -5 - 3

= -8

Part c

The value of f(0)

f(0) = -0 - 3

= -3

Therefore, the value of f(-4), f(5) and f(0) are 1, -8 and -3 respectively

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The two unit vectors parallel to the line y=5x+5 are [tex]u_{+}[/tex] =⟨1, 5⟩/√26 and [tex]u_{-}[/tex] =⟨1, 5⟩/√26.

Vectors, in Maths, are objects which have both, magnitude and direction. Magnitude defines the size of the vector. It is represented by a line with an arrow, where the length of the line is the magnitude of the vector and the arrow shows the direction.

Part A:

The given equation of line is y=5x+5.

It's all about slope. The vectors parallel to the line must have the same slope as the line. The line here is y=5x+5., and has slope 5. We can say the rise is 5 and the run is 1 and write the vector.

[tex]u_{\pm}[/tex] =⟨1, 5⟩

Magnitude =|u| =√(1²+5²)

= |±√26|

= √26

So, the two unit vectors are

[tex]u_{+}[/tex] =⟨1, 5⟩/√26 and [tex]u_{-}[/tex] =⟨1, 5⟩/√26

Part B: 2x+4y=1

x+2y=1/2

y=-x/2+1/4

Here slope is -1/2, where rise is -1 and run is 2

[tex]u_{+}[/tex]= ⟨2, -1⟩ and [tex]u_{-}[/tex]= ⟨-2, 1⟩

Magnitude =|u| =√(2²+(-1)²)

= √5

So, the two unit vectors are

[tex]u_{+}[/tex] =⟨2, -1⟩/√5 and [tex]u_{-}[/tex] =⟨-2, 1⟩/√5

Part C: The given equation is y=2-5x

Here slope is -5, slope perpendicular to line is -1/5

So, rise is -1 and run is 5

[tex]u_{\pm}[/tex] =⟨5, -1⟩

Magnitude =|u| =√(5²+(-1)²)

= √26

So, the two unit vectors are

[tex]u_{+}[/tex] =⟨5, -1⟩/√26 and [tex]u_{-}[/tex] =⟨-5, 1⟩/√26

Therefore, the two unit vectors parallel to the line y=5x+5 are [tex]u_{+}[/tex] =⟨1, 5⟩/√26 and [tex]u_{-}[/tex] =⟨1, 5⟩/√26.

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∠4=∠6 because each pair of alternate interior angles is equal ( The two column proof is attached ).

Every two-column proof has exactly two columns. One column represents our statements or conclusions and the other lists our reasons. In other words, the left-hand side represents our “if-then” statements, and the right-hand side explains why we know what we know.

We can clearly observe here that ∠1=∠5 , ∠2=6, ∠3=∠7, ∠4=∠8 as Corresponding angles are equal.

∠4 + ∠3 = 180°……..(1) PQ is the straight line

∠6 + ∠7 = 180°……..(2) t is the straight line

therefore from the two equations we get

∠4 =∠6

Hence, ∠4  ≅∠6 ( The two column proof is attached )

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The construction that the figure represents is perpendicular bisector

A perpendicular bisector is a line that cuts through the intersection of two other line segments at a right angle.

So, a perpendicular bisector always splits a line segment through its middle, as we can see. By definition, the word "bisect" denotes an even or uniform division.

Perpendicular Bisector Properties for AB in the figure

It cuts AB in half, or into two equal halves.

It is perpendicular to AB or at right angles to it.

The distance between each point along the perpendicular bisector from points A and B is equal.

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The true statements are as follows

This is a chi-square test for association

The null and alternative hypothesis is

The null hypothesis is that here is no association between state and sporting preference.

The alternative hypothesis is that there is an association between state and sporting preference.

as given in the question

a survey was conducted with large group of teenagers from

6 different states.

let the portion of teenagers who selected the sports most preferred as (p)

(p1) as the the teenagers that for state 1

(p2) as for the teenagers that for state 2

(p3) as for the teenagers that for state 3

(p4) as for the teenagers that for state 4

(p5) as for the teenagers that for state 5

(p6) as for the teenagers that for stage 6

the formula of chi-square test is

[tex]\chi^2=\sum\frac{(O_i-E_i)^2}{E_i}[/tex]

where

[tex]O_i[/tex] is the observed value

[tex]E_i[/tex] is the expected value

[tex]\chi^2[/tex] is chi squared value

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Note:

The full question is

A survey was conducted with a large group of teenagers from 6 different states and it asked them for their favorite sport out of a large list of sports. The study aims to look at whether a teenager's state of residence is associated with their sporting preferences.

Assume that p is the proportion of teenagers who selected the sport most preferred in their state of residence, such that p1 is that for State 1 , p2 is that for State 2 and so on.

Select from the following all statements that are true as well as the hypothesis that corresponds to this study:

[tex]This\ is\ a\ chi-square test for association.& \mathrm{H}_0: \mathrm{p}_1 \neq \mathrm{p}_2 \neq \ldots \neq \mathrm{p}_6 \\& \mathrm{H}_{\mathrm{a}}: \mathrm{p}_1 \neq \mathrm{p}_2 \neq \ldots \neq \mathrm{p}_6 \\& \mathrm{H}_0: \mathrm{p}_1=\mathrm{p}_2=\ldots=\mathrm{p}_6$\mathrm{H}_0$ : There is no association between state and sporting preference. $\mathrm{H}_{\mathrm{a}}$ : There is an association between state and sporting preference. This is a one-way ANOVA test for equality of means.[/tex]

[tex]$\mathrm{H}_0$ : There is an association between state and sporting preference.$\mathrm{H}_{\mathrm{a}}: \mathrm{p}_1=\mathrm{p}_2=\ldots=\mathrm{p}_6$$\mathrm{H}_{\mathrm{a}}$ : There is no association between state and sporting preference.[/tex]

On solving the question - At α = 0.05, two tailed critical value, Lower Bound =[tex]0.757[/tex], Upper Bound = 0.803, (0.757, 0.803)

An estimation range for an unknowable parameter is referred to as a confidence interval in frequentist statistics. The most popular confidence level is 95%, however other levels, such 90% or 99%, are occasionally used for computing confidence intervals. A constant that indicates the intended degree of significance based on the normal distribution is multiplied by the standard error once it has been calculated to get the confidence interval. Intervals with 95% certainty have a constant value of 1.96.

The number of people in the sample that felt protecting the rights of those with unpopular views is a very important component of a strong democracy is 1200*.78 = 936

Yes, the sample size is large enough, since the expected number of successes is 936, and the expected number of failures is 1200-936= 264, both of which are greater than or equal to 10.

95% Confidence interval :    

At α = 0.05, two tailed critical value

Lower Bound =[tex]0.757[/tex]

Upper Bound = 0.803

(0.757, 0.803)

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The correct equation for a1(t) is dt -4a1 + e4t a1(0) = 0.

An equation is a mathematical expression that shows the relationship between two values or variables. It typically consists of an equal sign (=), two expressions separated by an operator, and sometimes additional symbols like parentheses, fractions, or exponents. Equations are used to describe physical laws and conditions, solve problems, and make predictions. For example, the equation for the area of a rectangle is A = lw, where A is the area, l is the length, and w is the width.

This is because the eigenvalue of A which corresponds to the coefficient of a1(t) is 4, so the equation should contain e4t. The initial condition is a1(0) = 0, since the initial value of a1(t) is 0.

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In the proof the reason for (5)ΔVWX ≅ ΔXYZ  is  ASA , the correct option is (d) .

In the question ,

it is given that ,

Given: VW parallel to ZY  and WX = YX

to Prove: ΔVWX ≅ ΔXYZ .

the statement to prove ΔVWX ≅ ΔXYZ are given as

(1) VW parallel to ZY      ....given

(2) m∠W = m∠Y    ...alternate interior angle .

(3) WX = YX     ...given

(4) m∠VXW = m∠ZXY      .....vertically opposite angle

(5) ΔVWX ≅ ΔXYZ

In statement (5) , Since two angles and 1 side is included in proving the given two triangles congruent ,

the reason will be ASA congruence , that is option (d) .

Therefore , the reason for (5) is ASA .

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Answer:

[tex]x=-6[/tex]

Step-by-step explanation:

Solve for x.

Combine [tex]x[/tex] and [tex]\frac{1}{6}[/tex].

Combine [tex]x[/tex] and [tex]\frac{2}{3}[/tex].

[tex]-\frac{x}{6} -5=\frac{2x}{3}[/tex]

Move all terms containing x to the left side of the equation.

Subtract [tex]\frac{2x}{3}[/tex] from both sides of the equation.

[tex]-\frac{x}{6} -5-\frac{2x}{3}=0[/tex]

To write [tex]-\frac{2x}{3}[/tex] as a fraction with a common denominator, multiply by [tex]\frac{2}{2}[/tex].

[tex]-\frac{x}{6}-\frac{2x}{3}*\frac{2}{2}-5 =0[/tex]

Write each expression with a common denominator of 6, by multiplying each by an appropriate factor of 1.

[tex]-\frac{x}{6}-\frac{2x*2}{6}-5 =0[/tex]

Combine the numerators over the common denominator.

[tex]\frac{-x-2x*2}{6}-5 =0[/tex]

Simplify the numerator.

Factor [tex]x[/tex] out of [tex]-x-2x*2[/tex].

[tex]\frac{x(-1-2*2)}{6}-5 =0[/tex]

[tex]\frac{x(-1-4)}{6}-5 =0[/tex]

[tex]\frac{x*-5}{6}-5 =0[/tex]

Move the negative in front of the fraction.

[tex]-\frac{5x}{6} -5=0[/tex]

Add 5 to both sides of the equation.

[tex]-\frac{5x}{6}=5[/tex]

Multiply both sides of the equation by [tex]-\frac{6}{5}[/tex].

[tex]-\frac{6}{5}(-\frac{5x}{6})=-\frac{6}{5}*5[/tex]

Simplify the left side by cancelling the common factors of 5 and 6.

[tex]x=-\frac{6}{5} *5[/tex]

Move the leading negative in [tex]-\frac{6}{5}[/tex] into the numerator.

[tex]x=\frac{-6}{5} *5[/tex]

Cancel the common factor of 5.

[tex]x=-6[/tex]

A rating/review would be much appreciated. Wish you a happy holiday season!

Answer:

---------------------------------

Given equation:

Fist, multiply both sides by 6 to clear the fraction:

Collect the terms with the variable and solve for x:

On solving the provided question we can say that - mean of x mean of y

27.37136 30.73706

In statistical inference studies, an alternative hypothesis is a statement. This is represented by Ha or H1, which conflicts with the null hypothesis. It only serves as a zero replacement. The claims that researchers are investigating in hypothesis testing are alternative theories.

[tex]t = -11.067, \\df = 2301.5, \\p-value < 2.2e-16[/tex]

alternative hypothesis: true difference in means is less than 0

95 percent confidence interval:

[tex]-Inf -2.865266[/tex]

sample estimates:

mean of x mean of y

27.37136 30.73706

Conclusion:-

Here   p-value = 2.2e-16 < 0.05(Level of significance)

Therefore we reject  at 5% Loss

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Answer:

g = 18(1/3)

Step-by-step explanation:

The sub is made of 18 sections of 1/3, so it is 18(1/3) feet long (or 6ft when simplified)

Amount of money needed to carry to the sari sari store is 59.

a) According to question

Amount of eggs to buy = 3

Rate of eggs = 8

Total amount for eggs = 3x8=24

Amount of  hotdogs to buy = 2

Rate of hotdogs = 10

Total amount to pay for hotdogs = 2x10=20

Amount of  bread to buy = 3

Rate of bread = 5

Total amount to pay for bread=3x5=15

So money needed to take to the sari sari store=24+20+15=59

b) Polynomial is 4x²+2x-3

1) x=1

  4(1)²+2(1)-3

  4+2-3

  3

2) x=2

 4(2)²+2(2)-3

 4(4)+4-3

 16+4-3

 17

3) x=3

 4(3)²+2(3)-3

 4(9)+6-3

 36+6-3

 39

4) x=5

 4(5)²+2(5)-3

 4(25)+10-3

 100+10-3

 107

5) x=6

 4(6)²+2(6)-3

 4(36)+12-3

 144+12-3

 153

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The length of an interval of time such that the probability that no messages arrive during this interval is 0.94 is 56.88 seconds.

We have to find the length of the time interval that the probability of no messages arrive during this interval is 0.94

Let X be the no.of messages arrive to the server

X follows the poisson distribution with mean λ = 17

Let t be the time interval

The probability of no messages arrive during this interval is 0.94

Then P(0) = 0.94

e⁻⁽λt⁾ = 0.94

λt = ln0.94

17 t   = ln(0.94)

t = ln(0.94)/17

t = 0.026872 / 17 hrs

t = 0.0158

Convert 0.0158 hours to seconds

That is t=0.0158(60)(60)

 =56.88seconds

Hence in 56.88 seconds the time interval that the probability of no messages arrive during this interval is 0.94 .

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The Greatest Common Divisor (GCD) of 85 and 51 is 17.

The greatest common factor that divides two or more numbers is known as the greatest common divisor (GCD). The highest common factor is another name for it (HCF).

Explanation

Step 1: Divide 85 by 51. The quotient is 1 and the remainder is 34.

Step 2: Divide 51 by 34. The quotient is 1 and the remainder is 17.

Step 3: Divide 34 by 17. The quotient is 2 and the remainder is 0.

Step 4: Since the remainder is 0, the Greatest Common Divisor (GCD) of 85 and 51 is 17.

The GCD of 85 and 51 is 17.

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Preferred shares = 0.40/25.00 = 1.60%    

Common shares = 0.07/17.50 = 0.40%

What is stock  market?

The phrase "stock market" describes a number of marketplaces where shares of publicly traded firms can be purchased and sold. Such financial transactions take place on official exchanges and in over-the-counter (OTC) markets that adhere to a predetermined set of rules.

Year Total         Preference   Common   Dividend per

             dividend     dividend      dividend    common share

1       80000 80000 0 0.32 0.00

2       90000 90000 0 0.36 0.00

3       150000 130000 20000 0.52 0.04

4       150000 100000 50000 0.40 0.10

5       160000 100000 60000 0.40 0.12

6       180000 100000 80000 0.40 0.16

Note:       2.40 0.42

Preference dividend per year = 250000*$20*2%=  100000  

Preference dividend for the 3rd year includes arrear dividend of $20000 for the 1st year

and $10000 for the second year, in addition to the $100000 dividend of that year.

Average annual dividend per share:      

Preferred shares = 2.40/6 = $0.40      

Common shares = 0.42/6 = $0.07      

Average annual percentage return:      

Preferred shares = 0.40/25.00 = 1.60%    

Common shares = 0.07/17.50 = 0.40%

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