find the dimensions of the closed rectangular box with square base and volume 8000 cubic centimeters that can be constructed with the least amount of material.

2x+y-3 = 0 iS a linear function with coefficient as 2 and 1 with constant part c = 3.

In a graph, a linear function is one that generates a straight line. The degree of the polynomial function is often either maximum 1 or minimum 0. Despite the fact that both calculus and linear algebra can be used to express linear functions. The function notation is the only thing that differs. Also required is the ability to read an ordered pair expressed in function notation. A function is denoted by the notation f(a), where an is an independent variable on which the function depends. The straight line on the linear function graph contains the expression or formula;

Linear function can be always represented in the format of ax + by + c = 0.

Here,

a € ( -∞, +∞)

b € ( -∞, +∞)

c € ( -∞, +∞)

Hence, Linear expression always contains y¹ and x¹.

and can always be represented in the form of y = mx.

So it always represents the straight line.

For example: 2x+y-3 = 0 ia a linear expression with coefficient as 2 and 1 with constant part c = 3.

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Answer:

z=1

Step-by-step explanation:

2.2(z+3)-1.93=6.87

2.2z+6.6-1.93=6.87

2.2z+4.67=6.87

       -4.67. -4.67

2.2z=2.2

/2.2. /2.2

z=1

Hopes this helps please mark brainliest

Answer:

z = 1

Step-by-step explanation:

2.2(z + 3) − 1.93 = 6.87

1. Distribute

2.2z + 6.6 - 1.93 = 6.87

2. Combine Like Terms

2.2z + 4.67 = 6.87

3. Subtract 4.67 on both sides

2.2z + 4.67 = 6.87

        - 4.67  - 4.67

2.2z = 2.2

4. Divide both sides by 2.2

[tex]\frac{2.2z}{2.2} = \frac{2.2}{2.2}[/tex]

z = 1

Answer:

(8,2), (12,2),(10,10)

Step-by-step explanation:

any coordinate that means the two sides will have the same length

first, we know that the length is twice the amount of the width

so any coordinate we pick must be twice the length

pick a coordinate that is quite high up ( so we don't get into negatives)

so let's say (10,10)

then we want to go a positive number down - don't want to deal with fractions.

so e.g 8

so 6 down would be (10,2)

then the width is half the length so it has a 4cm wide base

and put an equal amount on each side to the left and the right to get the base co-ordinates

to the left : (8,2)
plus to the right: (12,2)

2+2= make a 4cm base

put these coordinates together and presto! you have coordinated proofs!

Brackets go first, then you have to do the multiplication and division, then addition and subtraction:

-42/5÷(-5) = (-42/5)÷(-5) = -42/(5 x -5) = -42/(-25) = 42/25

It has 816 ways to 3-element subsets, such that no two consecutive integers are in the subset.

You need three digits from 1 to 20, but you don't want any consecutive ones. You should call them a, b, and c.

Without loss of generality, a < b < c.

We use combinations,

Remember that we can write a + 1 b if a b and they're not sequential. With the aid of this insight, we can see the number of desired the number of methods for choosing a, b, and c from among the integers so that,

(1 ≤ a) & (a + 1 < b) & (b + 1 < c) & (c ≤ 20)

In other words, you want to know how many possibilities there are to choose the three numbers a, b, and c so that;

1 ≤ a < b − 1 < c − 2 ≤ 18

As a result, the question is equal to asking how many ways there are to select three integers (a, b - 1, and c - 2) from a range of 1 to 18. In other words, we are interested in how many ways we can select three items from a total of 18 options.

So, the response is:

[tex](^1^8_3)[/tex] = 816 ways

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Lacy's blue fish tank has now 3 2/3 gallons of water.

If Lacy has 10 gallons of water and she pour an equal number of gallons of water into each of her 3 fish tanks, then the amount of water each tank will have is equal to the quotient of 10 divided by 3.

amount of water poured in each tank =  10 gallons / 3

amount of water poured in each tank = 10/3 gallons

If Lacy's blue fish tank already has 1/3 gallon of water in it, then the total amount of water the blue fish tank has is the sum of the two fractions.

Add the initial amount and the amount poured in after.

1/3 + 10/3 = 11/3 =  3 2/3 gallons

Simplifying the fraction, Lacy's blue fish tank has now 3 2/3 gallons of water.

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The line segment i rotated 270 degree clockwise about the origin to form P'Q' the statement describe P'Q'. The accurate alternative is (B) P'Q' is same in period to PQ.

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Answer:

Step-by-step explanation:

8.5

The bearing of A route from C is  67.2°.

What is an angle?

An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which means "corner," is the source of the English term "angle." The shared terminus of two rays is known as the vertex, and the two rays are referred to as sides of an angle.

Here, we have

B traveled on a bearing of 150 and A traveled on the north side of B.

To find angle C, we have to use the cosine rule and we get

12² = 8² + 10² - 2 × 8 × 10 × cosC

C = 82.8°

The bearing of A from C will be

= 150° - 82.8°

= 67.2°.

Hence, the bearing of the A route from C is  67.2°.

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Hence, we proved that if n is an integer, then either n= 2mod3 or n^2=n mod6.

In the given question we have to prove that prove if n is an integer, then either n= 2mod3 or n^2=n mod6.

If n=2mod3 then tere is nothing to prove.

If n≠ 2mod3 then either n=0mod3 or n=1mod3

Case (1)

n=0mod3 ⇒ 3 | n⇒n=3m for mϵZ

n^2-n=(3m)^2-3m

n^2-n=9m^2-3m

n^2-n=3m(3m-1)

So, 3|3m, in this case 3m is even or 3m-1 is even.

i.e. 2|3m(3m-1) and also 3|3m(3m-1)

⇒6|3m(3m-1)

⇒6|n(n-1)

⇒6|(n^2-n)

⇒n=n mod6 (Proved)

Case (2)

n=1mod3 ⇒ 3 | (n-1)⇒n=3m+1 for mϵZ

n^2-n=(3m+1)^2-(3m+1)

n^2-n=9m^2+6m+1-3m-1

n^2-n=9m^2+3m

n^2-n=3m(m+1)

So, 3|3m, in this case 3m is even or 3m+1 is even.

i.e. 2|3m(3m+1) and also 3|3m(3m+1)

⇒6|3m(3m+1)

⇒6|n(n-1)

⇒6|(n^2-n)

⇒n=n mod6 (Proved)

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The write question is

Prove that if n is an integer, then n = 2 mod 3 or n^2 = n mod 6.

If the firm increases its production and sales from 3 to 4 units per day, the profit margin increases.

The correct answer option is option B

Total profit = Total revenue - Total cost

Total revenue = number of units × price of each Units

At 3 units:

Total profit = Total revenue - Total cost

= (3 × 2500) - $3000

= $7,500 - $3000

Total profit = $4,500

At 4 units:

Total profit = Total revenue - Total cost

= (4 × $2,500) - $4,100

= $10,000 - $4,100

= $5,900

In conclusion, the profit margin when production is increased from 3 to 4 units will also increase.

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The depth of the juice in the glass is 5.30cm

Given in the question,

length of juice box = 5cm

width of juice box = 3cm

Height of juice box = 10cm

Volume of juice box = volume of cuboid = length× width× height

= (5)(3)(10)

=(15)(10)

= 150 cm³

This juice then poured into a glass with radius of 3 cm

Glass is of cylindric shape

So, volume of cylinder = πr²h

= π(3)²h

Volume of juice should be equal to volume of glass

=>  π(3)²h = 150 cm³

=> (3.14)(9)h=150

=> h = 150 / (3.14)(9)

h = 150/28.26

h = 5.30cm

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By using the z value, it can be calculated that

P value for upper tail = 0.0455

P value for lower tail = 0.9545

P value for two tail  = 0.091

What is z value?

z value determines the number of standard deviation, the event is above the mean value. That means z value determines the distance of the event from the mean and it is measured in terms of Standard Deviation.

z value = 1.76

From the z table

P value for upper tail = 1 - 0.9545 = 0.0455

P value for lower tail = 0.9545

P value for two tail = 0.0455 [tex]\times[/tex] 2 = 0.091

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Answer:

y=-x+6

Step-by-step explanation:

For linear graphs, the slope of perpendicular lines must be of equal value but opposite in sign: ([tex]m_1=-m_2[/tex]).

The slope of y=x+1 is 1, therefore the slope of the perpendicular line must be -1.

Given (x,y)=(-2,8) the equation of the perpendicular line can be obtained by substituting these values into the slope-intercept form y=mx+b.

8=(-1)(-2)+b

8=2+b

6=b

So, the perpendicular line has a slope-intercept form of y=-x+6 or y=6-x.

The number of ways to choosing coins with at least 15 quarters and 20 dimes is 4354.

In the given question, if the pile contains only 15 quarters and only 20 dimes but at least 30 of each other kind of coin, then we have to find the collections of 30 coins can be chosen.

Number of coins only of quarters = 15

Number of coins only of dimes = 20

The number of ways to choosing coins with at least 15 quarters and 20 dimes= Total- [tex](^{18}C_{3}+^{13}C_{3})[/tex]

The number of ways to choosing coins with at least 15 quarters and 20 dimes= [tex]^{33}C_{3}-(^{18}C_{3}+^{13}C_{3})[/tex]

Using [tex]^{n}C_{r}=\frac{n!}{r!(n-r)!}[/tex]

The number of ways to choosing coins with at least 15 quarters and 20 dimes= [tex]\frac{33!}{3!(33-3)!}-(\frac{18!}{3!(18-3)!}+\frac{13!}{3!(13-3)!})[/tex]

The number of ways to choosing coins with at least 15 quarters and 20 dimes= [tex]\frac{33!}{3!30!}-(\frac{18!}{3!15!}+\frac{13!}{3!10!})[/tex]

Simplifying

The number of ways to choosing coins with at least 15 quarters and 20 dimes= [tex]\frac{33\times32\times31\times30!}{3\times2\times1\times30!}-(\frac{18\times17\times16\times15!}{3\times2\times1\times15!}+\frac{13\times12\times11\times10!}{3\times2\times1\times10!})[/tex]

The number of ways to choosing coins with at least 15 quarters and 20 dimes= 11×16×31-(3×17×16+13×2×11)

The number of ways to choosing coins with at least 15 quarters and 20 dimes= 5456-(816+286)

The number of ways to choosing coins with at least 15 quarters and 20 dimes= 5456-1102

The number of ways to choosing coins with at least 15 quarters and 20 dimes= 4354

If the pile contains only 15 quarters and only 20 dimes but at least 30 of each other kind of coin, the number of ways to choosing coins with at least 15 quarters and 20 dimes is 4354.

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As a seismologist, Janet understands the devastating effects that earthquakes can have. so, she keeps her basement stocked with canned food, like chili. This table shows the relationship between the number of cans of chili Janet buys, x, and the total cost (in dollars), y.

[tex]\begin{tabular}{|c|c|c|}\underline{x(cans)}&\underline{y(dollars)} \\12 & \$\ 26.40 \\ 44 & \$\ 56.80 \\ 66 & \$\ 77.70 \\ 100 & \$\ 110 \end{align}[/tex]

According to the values in the table, x and y have not a proportional relationship.

Table

[tex]\begin{tabular}{|c|c|c|}\underline{x(cans)}&\underline{y(dollars)} \\12 & \$\ 26.40 \\ 44 & \$\ 56.80 \\ 66 & \$\ 77.70 \\ 100 & \$\ 110 \end{align}[/tex]

Specifies the gradient (m) in rows 1 and 2

m₁ = [tex]\displaystyle \frac{56.80 - 26.40}{44-12}[/tex]

m₁ = [tex]\displaystyle \frac{30.40}{32}[/tex]

m₁ = [tex]\displaystyle \frac{19}{20}[/tex]

Specifies the gradient (m) in rows 2 and 3

m₂ = [tex]\displaystyle \frac{77.70 - 56.80}{66-44}[/tex]

m₂ = [tex]\displaystyle \frac{20.9}{22}[/tex]

Because the values of m₁ and m₂ are not the same [tex]\displaystyle \frac{19}{20}[/tex] ≠ [tex]\displaystyle \frac{20.9}{22}[/tex], then x and y have not a proportional relationship.

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The storage bin can hold 5477 bushels.

We have total capacity of the storage bin which is given to us in mixed fraction form.

If we convert this into improper fraction, it is equivalent to =27385/4

Now, in the same way ,we have also given that  some amount of wheat already fills which is also given in mixed fraction form.

If we convert this into improper fraction form, it is equivalent to

=5/4

Now, to find the the number of bushels storage bin maximally hold is =(total volume of storage bin)/(Volume of wheat fills)

=>N = (27385/4)/(5/4)

=>N=27385/5

=>N=5477

Hence, total number of bushels which storage bin can hold is 5477.

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We can determine which table represents a function by, verifying each input value's pairing with a single output value.

A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain.

Given,

We can determine which table represents a function by,

Determine whether a table of input and output values constitutes a function. Determine the values for the input and output. Verify each input value's pairing with a single output value. In that case, the table is a function.

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Answer:

$14

Step-by-step explanation:

since there are 2 halves in a pound we just need to multiply 7 by 2

7x2

14

Hopes this helps please mark brainliest

It's $7 for 1/2 a pound so we have to double it to get our answer so...

$14 would be our answer.

[tex]1/2+1/2=1[/tex]

If the data collection contains the following numbers: 3, 19, 25, 32, 36, 38, 31, 42, 57, 45, and 85 , Here 3 and 85 represents the outlier.

The important characteristics of Statistics are given as follows:

Outliers are the data's most wildly divergent values.

For instance, if the data collection contains the following numbers: 3 19, 25, 32, 36, 38, 31, 42, 57, 45, and 85

The display of data and outliers

The numbers 3 and 85, which are at the extremes and are hence outliers in the number line above, may be seen to be at these extremes.

The group's outliers are a part of it, but they are not close to the other group members.

Outliers are a concern since they cause an imbalance in the data set, which is why they are typically eliminated. Furthermore, an error can sometimes cause an outlier to appear in a data collection.

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The equation of the line that passes through the point (8,2) and is parallel to the line x+4y = 28 is y = -x/4+4.

According to the question,

We have the following information;

Point thorough which line is passing = (8,2)

Equation of the line:

x+4y = 28

Now, we will convert this equation into the slope-intercept form:

4y = -x+28

Dividing by 4 on both sides of the equation:

y = -x/4+28/4

y = -x/4+7

Slope of the line = -1/4

We know that the following formula is used to find the equation of the line:

(y-y') = m(x-x')

(y-2) = -1/4(x-8)

y-2 = -x/4+2

Adding 2 on both sides:

y = -x/4+4

Hence, the equation of the line that passes through the point (8,2) and is parallel to the line x+4y = 28 is y = -x/4+4.

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The probability on a particular day that all machines will break down is 0.00060840. So the option b is correct.

In the given question,

A company has four photocopy machines A, B, C and D.

The probability that a given machine will break down on a particular day is;

P(A) = 13⁄100

P(B) = 9⁄50

P(C) = 1⁄5

P(D) = 13⁄100

We have to find the probability on a particular day that all machines will break down.

The required probability = P(A∩B∩C∩D)

The required probability = P(A)∙P(B)∙P(C)∙P(D)

The required probability = 13⁄100 × 9⁄50 × 1⁄5 × 13⁄100

The required probability = 1521/2500000

The required probability = 0.00060840

Hence, the probability on a particular day that all machines will break down is 0.00060840. So the option b is correct.

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If p = 6, he will not meet his goal.

If p = 7, he will meet his goal.

If p = 8, he will meet his goal.

What is inequality?

In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to one another.

Here, we have

Jerome has already read 50 pages of a novel and read p pages each day from Monday to Friday.

If Jerome read at least 85 pages by the end of Friday, inequality will be,

50 + 5p ≥ 85

If Jerome reads 6 pages each day,

If p = 6,

Inequality will be,

50 + 5(6) ≥ 85

80 ≥ 85

It's not true.

Hence, he will not meet his goal.

If p = 7,

Inequality will be,

50 + 5(7) ≥ 85

85 ≥ 85

It's true.

Hence, he will meet his goal.

If p = 8,

Inequality will be,

50 + 5(8) ≥ 85

90 ≥ 85

It's true.

Hence, he will meet his goal.

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The equations of the electric curve using the coordinates x₁ and x₂ are:

For AB:

y₁= (1/EI) [-M₀x₁³/6L  + M₀ Lx₁/6]

For CB:

y₂= (1/EI) [-M₀x₂²/2   + 3M₀Lx₂/2  - M₀L]

The elastic curve of a beam is the curve formed by the intersection of the neutral surface and the side of the beam when the longitudinal stresses on the fibers.

An Elastic curve is flatter, resembling the horizontal lines in the letter E. Price elasticity of demand, also known as demand elasticity, relates to the degree of responsiveness in demand quantity with respect to price.

For AB:

[tex]M_{x}[/tex]= - [tex]A_{y}[/tex]x₁

By using double integration method,

EI y₁"=[tex]M_{x}[/tex]= - [tex]A_{y}[/tex]x₁

EI y₁'=- [tex]A_{y}[/tex](x₁)²/2   +C₁   -------(1)

EI y₁=- [tex]A_{y}[/tex](x³/6)+C₁ x +C₂   -------(2)

The boundary conditions are:

At x₁=0, y₁=0

At x₁=L, y₁=L

y₁=0,

=- [tex]A_{y}[/tex](0)+C₁ (0) +C₂

Then, C₂=0

y₁=L,

=- [tex]A_{y}[/tex](L³/6)+C₁ (L) +0

Then, C₁= [tex]A_{y}[/tex](L²/6)

C₁=M₀L/6

EIy₁'= -M₀L((x₁)²/2) ------(3)

EIy₁ = -M₀L((x₁)³/6)+(M₀L/6)x₁   -----(4)

For CB:

Mₓ= -M₀

x₂ is in opposite direction to x₁

So the integration with respect to x₂ will be alternately signed.

EI y₂"=Mₓ= -M₀

EI y₂'= -(-M₀x₂)+C₃

EI y₂'=M₀x₂+C₃ ------(5)

EI y₂= -M₀((x₂)²/2)-C₃x₂+C₄ ------(6)

The boundary conditions are:

x₂=L, y₂'=y₁'

y₂=y₁=0

At x₂=L,

y₂'= M₀L²+C₃

y₂'= y₁'= -M₀L/2

C₃= -3M₀L/2

y₂= -M₀L₂/2-(-3M₀L/2)+C₄

C₄= -M₀L

The equation of the electric curve is,

For AB:

y₁= (1/EI) [-M₀x₁³/6L  + M₀ Lx₁/6]

For CB:

y₂= (1/EI) [-M₀x₂²/2   + 3M₀Lx₂/2  - M₀L]

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The sum of given equation is 96/13.

Given equation is sum from n equals 1 to infinity of 12 times negative five eighths end quantity to the power of n that means

∑ n=1 to ∞ [12(-5/8)ⁿ]

This is in the form of geometric expression

that is an = ar^(n-1)

therefore we get

a = 12 and r = - 5/8

We have to find the sum.

Sum of infinity of the sequence is expressed as

Sum = a / (1-r)

where a  = first term of the series

and r = common ratio between two consecutive terms

Now we will substitute values of a and r in the above equation

Sum = [tex]\frac{12}{1-(-5/8)}[/tex]

        = [tex]\frac{12}{1 + 5/8}[/tex]

        = [tex]\frac{12}{13/8}[/tex]

Sum = [tex]\frac{96}{13}[/tex]

Hence the sum of given equation is 96/13.

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The required radius of given circle is 30 units.

A circle is a shut two-layered figure wherein the arrangement of the multitude of focuses in the plane is equidistant from a given point called "focus". Each line that goes through the circle shapes the line of reflection evenness. Likewise, it has rotational evenness around the middle for each point.

Picture result for What is circle and region of its area?

The equation for the region of the area of a circle is(r2) /360where r is the span of the circle and is the point of the area.

Angle of sector = 100 degree, area= 250π

Let radius of circle be r

Then,

area of sector = (r^2) /360

250π =  (r2) 100/360

r^2 = 250×360/100

r = √25×36 = 30 units

Thus, radius of the circle is 30units.

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These are the preposition for the given conditions:

(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).

The domain of the propositional P(x, y) consists of pairs x and y, where x is 1, 2, 3 and y is 1, 2, 3.

What are the disjunction and conjunction?

In logic functions, there are two kinds of compound statements. These are referred to as conjunction and disjunction. A conjunction implies that both statements are correct, whereas a disjunction implies that at least one of the statements is correct. A conjunction connects statements with the word "and," whereas a disjunction connects statements with the word "or."

By using disjunction and conjunction we get,

a) ∀x∀y P(x, y) is written as:

(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).

b) ∃x ∃y P (x, y) is written as:

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

c) ∃x∀y P(x, y)  is written as:

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

d) ∀y∃x P(x, y)  is written as:

(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).

Hence, these are the preposition for the given conditions:

(P(1,1) ∧ P(1,2) ∧ P(1,3)) ∧ (P(2,1) ∧ P(2,2) ∧ P(2,3)) ∧ (P(3,1) ∧ P(3,2) ∧ P(3,3)).

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∨ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∨ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

(P(1,1) ∨ P(1,2) ∨ P(1,3)) ∧ (P(2,1) ∨ P(2,2) ∨ P(2,3)) ∧ (P(3,1) ∨ P(3,2) ∨ P(3,3)).

(P(1,1) ∧ P(1,2) ∧ P(1,3)) v (P(2,1) ∧ P(2,2) ∧ P(2,3)) v (P(3,1) ∧ P(3,2) ∧ P(3,3)).

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The allowable child and dependent care credit considering their pre-credit tax liability is $0

The following chart illustrates how their authorized child and dependent care credit was calculated:

If the revenue is less than $22,000, the full 22% would be permitted.

However, the eligible kid expense would only be allowed up to $7,000.

Consequently, the sum would be

= $7,000 × 22%

= $1,540

Since there is already a pre-credit tax liability, $0 should be taken into account because there would be no credit.

Due to their pre-credit tax liability, the available child and dependent care credit is therefore $0.

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By using the concept of probability, it can be calculated that

a) State space S(x) = {-1, 0, 1, 2}

b) The required probability mass function is

    P(X = -1) = [tex]\frac{125}{216}[/tex]

    P(X = 0)  [tex]= \frac{25}{72}[/tex]

    P(X = 1)  [tex]= \frac{5}{72}[/tex]

    P(X = 2)  [tex]= \frac{1}{216}[/tex]

c) Expected value of X  = -0.5

d) the stake of the gambler in  order to make this a fair game is $0.5

What is probability?

Probability gives us the information about how likely an event is going to occur

Probability is calculated by Number of favourable outcomes divided by the total number of outcomes.

Probality of any event is greater than or equal to zero and less than or equal to 1.

Probability of sure event is 1 and probability of unsure event is 0.

a) Let a number n be chosen between 1 and 6

Three fair six sided dice are thrown

The following cases can occur

1) Number n occur on none of the three dice

2) Number n occur on one dice

3) Number n occur on two dice

4) Number n occur on three dice

Let X denote your net win after one round of gambling (i.e., the payment received from the casino minus your stake)

1) X = 0  -1 = -1

2) X = 1 - 1 = 0

3) X = 2 - 1 = 1

4) X = 3 - 1 = 2

State space S(x) = {-1, 0, 1, 2}

b) P(X = -1) = [tex]{3 \choose 0}(\frac{1}{6})^0(\frac{5}{6})^3 = (\frac{5}{6})^3 = \frac{125}{216}[/tex]

    P(X = 0) = [tex]{3 \choose 1})(\frac{1}{6})(\frac{5}{6})^2 = \frac{25}{72}[/tex]

    P(X = 1) = [tex]{3 \choose 2}(\frac{1}{6})^2\frac{5}{6} = \frac{5}{72}[/tex]

    P(X = 2) = [tex]{3 \choose 3}(\frac{1}{6})^3(\frac{5}{6})^0 = \frac{1}{216}[/tex]

    The required probability mass function is

    P(X = -1) = [tex]\frac{125}{216}[/tex]

    P(X = 0)  [tex]= \frac{25}{72}[/tex]

    P(X = 1)  [tex]= \frac{5}{72}[/tex]

    P(X = 2)  [tex]= \frac{1}{216}[/tex]

c) Expected value of X = [tex]-1 \times \frac{125}{216} + 0 \times \frac{25}{72} + 1 \times \frac{5}{72} + 2\times \frac{1}{216}[/tex]

                                      = [tex]\frac{-125 + 0 + 15 +2}{216}\\[/tex]

                                      = [tex]-\frac{108}{125}[/tex]

                                      = -0.5

d) The gambler looses $0.5 on a average

   So this is not a fair game

   Let the stake of the gambler in  order to make this a fair game be t

   By the problem,

   -0.5 + t = 0

     t = 0 + 0.5

     t = $0.5

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