Kristen works for Lincoln Canning Company. She wants to make labels for several cans like the one shown below, only covering the lateral surface area. She is going to cut them from a sheet of plastic film that has an area of 2,265 square inches.

6 inches . 5 inches

What is the maximum number of labels she could make assuming there is minimal film wasted?

The correct selling price of an item will be $12.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that a  store is instructed by corporate headquarters to put a markup of ​25% on all items. An item costing ​$24 is displayed by the store manager at a selling price of ​$6.

The selling price will be calculated as:-

Selling price = 25% of 24 + $6

Selling price = 6 + 6

Selling price = $12

Hence, the selling price of the item is $12.

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The graph of the function is V-shaped, vertex domain and range are (-3,0), all the real values and [0, ∞);

The absolute value function, with vertex (h,k), is defined as follows:

y = |x - h| + k.

The features of the function are given as follows;

Vertex: (h,k).

Domain: All real values;

Range: [k, ∞).

Given;

g(x) = |× + 31|

Therefore, for the first item of the function is |x + 3|, since we just have to identify the features;

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The slope of the graph of the equation: y = 2x + 5

Slope = 2

The y-intercept of the graph of the equation: y = 2x + 5

Y-intercept = 5

The slope of the graph of the equation: y = 5 - 3x

Slope = -3.

The y-intercept of the graph of the equation: y = 5 - 3x

Y-intercept = 5

An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.

The general form of the equation of the line:-

y = mx + c

m = slope

c = y-intercept

Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )

1) y = 2x + 5 Slope = 2 is the slope of the equation's graph.

2.) The equation's graph's y-intercept is written as y = 2x + 5 Y-intercept = 5.

3) y = 5 - 3x Slope = -3 is the slope of the equation's graph.

4) The graph's y-intercept is as follows: y = 5 - 3x Y-intercept = 5.

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Compared to your regular wage without overtime, this would push your monthly budget up by $607.63.

In a budget, projected expenses are stated together with suggestions for how to fund them. A budget can show a surplus of resources for future use or a deficit where expenses are greater than income or other resources.

Surplus Budget, Balanced Budget, and Deficit Budget are 3 types of annual governmental budgets based on estimation. A balanced budget is one in which the revenues are larger than the expenses or equal to them. In the above link, you can read about the Highlights of the Union Budget for UPSC.

Let's calculate the regular earnings for 40 hours of work first:

Regular earnings = $11.75 x 40 hours = $470

Next, let's calculate the overtime earnings:

Overtime rate = $11.75 x 1.5 = $17.63

Overtime earnings = $17.63 x 20 hours = $352.60

Now let's add the regular earnings and overtime earnings to get the total gross pay for the month:

Total gross pay = $470 + $352.60 = $822.60

Now let's calculate the deductions:

FICA = $822.60 x 7.65% = $63.17

Federal tax withholding = $822.60 x 10.75% = $89.11

State tax withholding = $822.60 x 7.5% = $61.69

Total deductions = $63.17 + $89.11 + $61.69 = $214.97

So, the net pay for the month (total gross pay minus total deductions) would be:

Net pay = $822.60 - $214.97 = $607.63

Therefore, working 20 hours of overtime would increase your budget by $607.63.

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Answer:

Step-by-step explanation:

You can solve for a, b, and c by using a system of linear equations.

Starting with the first equation:

a + b + c = 90

You can isolate a by subtracting b and c:

a = 90 - b - c

Next, substitute the expression for a into the second equation:

12(90 - b - c) + 10b + 6c = 870

Expanding the equation:

1080 - 12b - 12c + 10b + 6c = 870

Combining like terms:

1070 - 2b + -6c = 870

Adding 2b and 6c to both sides:

1070 = 870 + 2b + 6c

Subtracting 870 from both sides:

200 = 2b + 6c

Dividing both sides by 2:

100 = b + 3c

Finally, subtracting 3c from both sides:

100 - 3c = b

To find c, substitute the expressions for a and b back into the first equation:

a + b + c = 90

90 - b - c + b + c = 90

90 - c = 90

-c = 0

Therefore, c = 0.

To find b, substitute c = 0 into the expression for b:

100 - 3c = b

100 = b

Therefore, b = 100.

Finally, to find a, substitute b = 100 and c = 0 into the expression for a:

a = 90 - b - c

a = 90 - 100 - 0

a = -10

Note that negative values for a, b, and c are possible, but not necessarily meaningful for this specific problem or context.

The solution is:

a = -10

b = 100

c = 0

Answer:

Width=56 yards

length=224 yards

Step-by-step explanation:

let:

Width=x

length= x*4= 4x

perimeter = 2*x + 2*4x

working

[tex](x \times 2) + (4x \times 2) = 560[/tex]

[tex]2x + 8x = 560[/tex]

[tex]10x = 560[/tex]

[tex] \frac{10x}{10} = \frac{560}{10} [/tex]

[tex]x = 56[/tex]

[tex]4x = 56 \times 4 = 224[/tex]

width =56 yards

length = 224 yards

The minimum value of [tex]$\|\mathbf{v}\|$[/tex] is zero.

The minimum value of [tex]$\|\mathbf{v}\|$[/tex] is zero, as it is the distance between the origin and a point on the line. To solve this problem, the formula [tex]$\|\mathbf{v}\| = \sqrt{x^2 + y^2}$[/tex] can be used, where x and y are the coordinates of the point on the line. Since the equation of the line is [tex]$\dbinom{10}{3} \dbinom{-2}{1}t$[/tex], the coordinates of the point on the line can be found by substituting [tex]$t = 0$: $x = 10 \cdot 0 - 2 \cdot 0 = 0$, and $y = 3 \cdot 0 + 1 \cdot 0 = 0$[/tex] . Substituting these values into the formula for [tex]$\|\mathbf{v}\|$ gives $\|\mathbf{v}\| = \sqrt{0^2 + 0^2} = 0$[/tex]. Therefore, the minimum value of [tex]$\|\mathbf{v}\|$[/tex] is zero.

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(a) The probability that a call is classified as long (L) is [tex]$P[L] = 1 - P[B_3] = 1 - \alpha (1-\alpha)^2 = 0.829$.[/tex]

(b) The probability that a call will be billed for nine minutes or less is

[tex]P[B1] + P[B2] + P[B3] + P[B4] + P[B5] + P[B6] + P[B7] + P[B8] + P[B_9] = \alpha + \alpha(1-\alpha) + \alpha(1-\alpha)^2 + \alpha(1-\alpha)^3 + \alpha(1-\alpha)^4 + \alpha(1-\alpha)^5 + \alpha(1-\alpha)^6 + \alpha(1-\alpha)^7 + \alpha(1-\alpha)^8 = 0.958[/tex]

Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is expressed as a number from 0 to 1, with 0 indicating that an event is impossible and 1 indicating that it is certain to occur. Probability is used to calculate the chance of a random event happening or to predict the outcome of a situation. Probability theory is also used to describe the underlying mechanics and dynamics of random phenomena

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Complete question:

For the telephone usage model of Example 1.18, let [tex]$B_m$[/tex] denote the event that a call is billed for $m$ minutes. To generate a phone bill, observe the duration of the call in integer minutes (rounding up). Charge for M minutes [tex]$M=1,2,3, \ldots$[/tex] if the exact duration T is [tex]$M-1 < t \leq M$[/tex]. A more complete probability model shows that for $m=1,2, \ldots$ the probability of each event [tex]$B_m$[/tex] is

[tex]$$P\left[B_m\right]=\alpha(1-\alpha)^{m-1}$$[/tex]

where [tex]$\alpha=1-(0.57)^{1 / 3}=0.171$[/tex]

A. The pair of segments that is parallel will be AB & EF, DH & CG, and BC & FG. Then the correct option is A.

B. The line G and line K will be a skew line. Then the correct option is B.

One of the first areas of mathematics is geometry, along with maths. It is preoccupied with spatial characteristics including the separation, form, size, and relative placement of objects. A geometer is a mathematician who specializes in shape.

A.  The parallel edges of the rectangular is given below.

AB = CD = EF = GH

AE = DH = CG = BF

BC = AD = EH = FG

The pair of segments that is parallel will be AB & EF, DH & CG, and BC & FG. Then the correct option is A.

B.  The line G and line K will be a skew lines. Then the correct option is B.

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The exact values of the remaining trigonometric functions of θ are:

sin(θ) = √(1/64)

cos(θ) = -8

tan(θ) = -8√(1/64)

csc(θ) = 1

cot(θ) = -√(1/64) / 8

Given that sec(θ) = -8 and sin(θ) > 0, we can find the exact values of the remaining trigonometric functions using the Pythagorean identity:

sec^2(θ) = 1/sin^2(θ)

Substituting the value of sec(θ), we have:

(-8)^2 = 1/sin^2(θ)

64 = 1/sin^2(θ)

sin^2(θ) = 1/64

sin(θ) = ±√(1/64)

Since sin(θ) > 0, we take the positive square root:

sin(θ) = √(1/64)

Next, we use the reciprocal identity for cosecant:

csc(θ) = 1/sin(θ)

Substituting the value of sin(θ), we have:

csc(θ) = 1/√(1/64) = 8√(1/64) = 8/√(64) = 8/8 = 1

Next, we use the reciprocal identity for cotangent:

cot(θ) = 1/tan(θ)

We can find the value of tan(θ) using the definition:

tan(θ) = sin(θ) / cos(θ)

Substituting the values of sin(θ) and cos(θ), we have:

tan(θ) = √(1/64) / (-8) = -√(1/64) / 8

Finally, we use the reciprocal identity for a tangent:

tan(θ) = 1/cot(θ)

Substituting the value of cot(θ), we have:

tan(θ) = -8√(1/64)

Therefore, the exact values of the remaining trigonometric functions of θ are:

sin(θ) = √(1/64)

cos(θ) = -8

tan(θ) = -8√(1/64)

csc(θ) = 1

cot(θ) = -√(1/64) / 8

The complete question is:-

Find the exact value of each of the remaining trigonometric functions of

θ. Rationalize denominators when applicable.

secθ=−8, given that sinθ>0

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The sequence of the grains should be from smallest to largest as follows:

1. Breccia, Siltstone

2. Conglomerate

3. Sandstone

4. Shale

A sequence is an enumerated group of items in mathematics where repetitions are permitted and order is important.

It has members, just like a set does.

The length of the sequence is determined by the number of items.

You should be familiar with the following four main categories of sequences: arithmetic sequences, geometric sequences, quadratic sequences, and special sequences.

So, because the grain size of breccia and siltstone is the same, they should be assigned the same number.

- A cement-like substance holds angular shards of varying sizes together to form a rock called breccia.

- It can also be generated from volcanic ash or other materials but is often formed from angular shards of other rocks.

- A sedimentary rock called siltstone is made up of clay-sized particles and often has smaller grains than sandstone.

- A cementing substance holds rounded or sub-rounded grains and pebbles together to form the sedimentary rock known as a conglomerate.

- Sand-sized particles make up the sedimentary rock known as sandstone, which is often created from quartz grains.

- The last sedimentary rock is shale, which typically has a grain size lower than sandstone and is made up of clay-sized particles.

Therefore, the sequence of the grains should be from smallest to largest as follows:

1. Breccia, Siltstone

2. Conglomerate

3. Sandstone

4. Shale

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Complete question:

points question at position 6 put the following rocks into proper order, from smallest grains (1) to largest (4). note: there are a pair of rock types that have the same grain size. give them the same number.

A) Breccia

B) Conglomerate

C)Sandstone

D) Shale

E) siltstone

Yes, the function f(x) = 3 sin(2x) is periodic. The smallest positive period of this function is 2π/2 = π.

f(x) = 3 sin(2x) This is because the function is sinusoidal and the period of a sinusoidal function is 2π/n,

where n is the coefficient in front of the x (which in this case is 2). Thus, the period of f(x) = 3 sin(2x) is 2π/2 = π.

This means that the function will repeat itself after every π units. To verify this, we can plug in x = 0 and x = π into the equation and see that we get the same value, 3 sin(2(0)) = 3 sin(0) = 0 and 3 sin(2(π)) = 3 sin(2π) = 0, showing that the function repeats itself after every π units.

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Based on the provided information, standard error is 0.138, z-score is 2.898, and p value is 0.0013995.

The standard error of a statistic refers to the standard deviation of its sampling distribution or an estimate of that standard deviation. It indicates how different the population mean is likely to be from a sample mean. Standard error of two samples is given by:

SE = √((S1^2)/n1+(S2^2)/n2)

In the given case s1 = 0.5, n1 = 40, s2 = 0.8, and n2 = 50. Hence,

SE = √(〖0.5〗^2/40+〖0.8〗^2/50)=0.138

Z-score for two samples is given by:

z = ((x1-x2)/SE) where x1 and x2 are sample means

In the given case x1= 5.35 and x2 = 4.95. Hence,

z = ((x1-x2)/SE) = (5.35-4,95)/0.138=2.898

The p -value from the table is when p(x >z) = 0.0013995.

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Answer:

2560/11

Step-by-step explanation:

�

=

640

11

and

�

=

2560

11

The total time elapsed on the astronauts' chronometers is 20 years. This is calculated by subtracting the time spent traveling to and from the distant planet (2 * 10 years = 20 years) from the total time elapsed on Earth (26 years).

This result takes into account the time dilation effect of special relativity, which states that time passes more slowly for a moving object relative to a stationary observer. Since the astronauts were traveling at high speeds for much of their journey, the time elapsed on their chronometers would be less than the time elapsed on Earth.

The total time elapsed on the astronauts' chronometers is 20 years. This is calculated by subtracting the time spent traveling to and from the distant planet (2 * 10 years = 20 years) from the total time elapsed on Earth (26 years).

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The area of the right triangle is 80.5 square units.

The area of a triangle is described as the total region that is enclosed by the three sides of any particular triangle.

In the right triangle, one of the legs is equal to 7, as the side length of the square inscribed in it.

We will use the Pythagorean theorem to find the length of the other leg and the calculation is shown below:

a^2 + b^2 = c^2

7^2 + b^2 = 24^2

49 + b^2 = 576

b^2 = 527

b = √t(527) = 23

The area of the right triangle is then:

A = (1/2)ab = (1/2) * 7 * 23 = (1/2) * 161 = 80.5 square units

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The values of a and b, considering the range of the function, are given as follows:

The range of a function is the set that contains all the output values that can be assumed by the function.

Hence, considering the graph, the range of a function is given by the values of y from the graph.

The minimum and maximum value of the function are given as follows:

Hence the values of a and b are given as follows:

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The image of the point b(7/2 , -6/5) is (7/18,-2/15)

A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates,

Given here: A dilation has center (0,0) and image of the point b (7/2,-6/5)

Now it is given that the scale factor is 1/9

Thus the image of the given point is

x=7/2×1/9

x=7/18

y=-6/5×1/9

y=-2/15

Thus the image is (7/18,-2/15)

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The volume of the solid that is generated when the region that is enclosed by the given curves V=253/5π

When the volume of a solid is calculated by using the shell method, the result will be a definite integral that represents the sum of the volumes of infinite thick cylinders.

If the axis of rotation is horizontal, the cylinders are horizontal same. The volume of a horizontal cylinder will be 2πrh dy.

The graph defines the area enclosed by a parabola concave upwards, the y-axis, and the line x = 3 in the first quadrant.

There is the volume of the solid is calculated by using the formula:

[tex]V=2\pi \int\limits^d_c {y[g(y)-h(y)]} \, dy[/tex]

y is the radius of the cylinder

h is the height of the cylinder.

Given information is that:

g(y)=3, h(y)=y¹/², c=0, d=9

Now plug the values in the formula:

[tex]V=2\pi \int\limits^9_0 {y(3-y^\frac{1}{2} )} \, dy\\\\\\V=2\pi \int\limits^9_0 (3y-y^\frac{3}{2} )} \, dy[/tex]

Solving the integral:

[tex]V=2\pi (\frac{3}{2} y^2-\frac{2}{5} y^\frac{3}{5})|_0^9[/tex]

Apply the fundamental theorem of calculus:

[tex]V=2\pi [(\frac{3}{2} (9)^2-\frac{2}{5} (9)^\frac{3}{5})-V=2\pi (\frac{3}{2} (0)^2-\frac{2}{5} (0)^\frac{3}{5})]\\\\V=2\pi (\frac{243}{10} -0)\\\\V=2\pi (\frac{243}{10})\\[/tex]

V=243/5π

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the real limits are; Lower Real Limits = 249.5 and Upper Real Limits = 250.5

Given the data in the question;

1 minute being the smallest unit on the scale

so we say; 1/2 which 0.5 minutes which we will use to calculate pur real limits time;

given that; the first subject sleeps 250 minutes

so

Lower Real Limits will be;

Lower Real Limits will = 250 - 0.5 = 249.5

Also our Upper Real Limits will be;

Upper Real Limits = 250 + 0.5 = 250.5

Therefore; the real limits are;

Lower Real Limits = 249.5

Upper Real Limits = 250.5

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The range of possible numbers will be  (-∞ , 16.6] as per the given conditions.

In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.

Given, I’m thinking of a number. When you multiply it by 5 and subtract 3 you will get a number less than 80.

Let "x" be the number

Thus from the given expression

=> 5*x - 3 < 80

add 3 on both sides

=> 5x -3 + 3 < 80 +3

=> 5x < 83divide by 5 into both sides

=> x < 83/5

=> x < 16.6

thus the range of x for the given Condition will be (-∞ , 16.6]

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The cubic regression equation is y = 27.13x^3 - 473.44x^2 + 2829.33x - 4381.69 and the estimated the number of graduates in 2001 is 5565

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

See attachment

Using a graphing calculator, we have the following summary:

The model from the above summary is

y = 27.1349x^3 - 473.4406x^2 + 2829.3314x - 4381.6892

Approximate

y = 27.13x^3 - 473.44x^2 + 2829.33x - 4381.69

In the year 2001, we have

x = 11

So, we have

y = 27.13 * 11^3 - 473.44 * 11^2 + 2829.33 * 11 - 4381.69

Evaluate

y = 5565

The estimated the number of graduates in 2001 is 5565

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There are 280 students attending the play.

Each term in a linear equation has an exponent of 1, and when this algebraic equation is graphed, it always produces a straight line. It is called a "linear equation" for this reason.

Both linear equations with one variable and those with two variables exist.

Given  student tickets cost $6 each and adult tickets cost $11 each,

let a number of students be x and the number of tickets be y

total tickets sold = 400

x + y = 400

and total amount made = $3000

so 6x + 11y = 3000

from equation 1,

y = 400 - x

substitute in the equation,

6x + 11y = 3000

6x + 11(400 - x) = 3000

6x + 4400 - 11x = 3000

-5x = 3000 - 4400

-5x = -1400

x = 1400/5

x = 280

and y = 400 - 280

y = 120

Hence 280 students attended the play.

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The point that is a solution if the inequality:

y < (2/3)*x + 2

is (1, 2).

Here we want to find a solution for the inequality below:

y < (2/3)*x + 2

To check if an ordered pair is a solution, then we need to replace the values of the ordered point and check if the inequality is true.

If we try with the first point we will get:

3 < (2/3)*0 + 2

3 < 2

That is false.

The correct option is (1, 2)

2 < (2/3)*1 + 2

2 < 8/3

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1. An equation that represents "the amount of tax on an item of price p",

t = p x X / 100

2. Marta does pay on an item whose price is $20, $1.6

Percentage is a way to express a number as a fraction of 100. It is often used to represent ratios and proportions in a more convenient and understandable form, especially in financial and statistical contexts. For example, 50% means 50 per 100, or half of a given quantity. It is denoted using the symbol "%".

Given that,

Tax rate = 8%

1. Let's assume that the sales tax rate is X%.

Then, the equation that represents the amount of tax (t) Marta pays on an item whose price is p can be written as:

t = p x X / 100

2. If the price of the item is $20, the amount of tax Marta pays can be calculated as follows:

t = $20 x X / 100

t = $20 x 8 / 100

t = $1.6

So, Marta pays $1.6 in sales tax on an item whose price is $100.

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Answer:

1. Women

Store

66-34= 32

Online

13

Total

32+13= 45

2. Men

Total

34+9=43

3. Total

online

13+9=22

total

66+22= 88

The sample standard deviation is 0.03

As per the given data, a quality engineer has taken the 20 samples in which each containing of 100 batteries.

The total number of the defective batteries which are observed over the 20 samples is 200.

Here we have to calculate the value of the sample standard deviation

The number of samples, k = 20

The number of batteries in each sample:

The sample size = 100

The total number of batteries of the sample are:

The total number of samples, [tex]$\sum{n=20 \times 100=2000}$[/tex]

[tex]& \bar{n}=\sum_\frac{n}{k}} \\[/tex]

= [tex]$\frac{2000}{20}[/tex]

= 100

The total number of defective batteries = 200

Now we can calculate the proportion of the deflective battery:

Therefore, [tex]$\bar{p}=$[/tex] The total number of defectives ÷ The total number of samples

= [tex]$\frac{200}{2000}[/tex]

= 0.10

The standard deviation, [tex]$\sigma_{\mathrm{p}}=\sqrt{\frac{\bar{p} \bar{q}}{\bar{n}}}$[/tex]

= [tex]$ \sqrt{\frac{0.1 \times 0.9}{100}} \\[/tex]

= [tex]& \sqrt{0.0009} \\[/tex]

=  0.03

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When a journal entry contains a debit to inventory, a corresponding credit to an offsetting account, such as Cost of Goods Sold (COGS), is required.

A journal entry that contains a debit to inventory requires a corresponding credit to an offsetting account, such as Cost of Goods Sold (COGS). The formula for the entry would be:

Debit Inventory

Credit Cost of Goods Sold (COGS)

The calculation would depend on the cost of the inventory, the quantity purchased and the applicable tax rate. For example, if the cost of the inventory is $5,000, the quantity purchased is 500 and the applicable tax rate is 5%, the journal entry would look like this:

Debit Inventory $5,000

Credit Cost of Goods Sold (COGS) $4,750 ($5,000 x 95% = $4,750)

In this example, the total cost of the inventory was $5,000, but the credit to COGS was calculated by multiplying the cost of the inventory ($5,000) by 95% (the applicable tax rate of 5% was subtracted). The resulting amount of $4,750 was the amount credited to COGS.

In summary, when a journal entry contains a debit to inventory, a corresponding credit to an offsetting account, such as Cost of Goods Sold (COGS), is required. The formula for the entry is: Debit Inventory, Credit Cost of Goods Sold (COGS). The calculation for the entry depends on the cost of the inventory, the quantity purchased and the applicable tax rate.

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There are 435 members in the House of Representatives.

Algebraic expressions are mathematical statements with a minimum of two terms containing variables or numbers.

Let's call the number of members in the Senate "s".

The expression for the number of members in the House of Representatives is "h".

According to the problem, there are 35 more members in the House than four times the number of members in the Senate, so the expression for the number of members in the House is:

h = 4s + 35

If there are 100 members in the Senate, then we can substitute this value into the expression:

h = 4 × 100 + 35

h = 435

So, there are 435 members in the House of Representatives.

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P(A) is the probability of observing 3 on first trial. P(B) is probability of observing 3 after at least 2 trials. P(A U B) is total probability of observing 3.

(a) The probability of observing a 3 on the first trial is 1/6. So, p(a) = 1/6.

(b) The probability of not observing a 3 on the first trial is 5/6.

If a 3 is not observed on the first trial, then the same experiment will be repeated with a 5/6 chance of not observing a 3 again.

This can be represented as (5/6)^2.

The probability of not observing a 3 on the first two trials is (5/6)^2.

So, the probability of observing a 3 on the third trial is

1 - (5/6)^2

This can be represented as

1 - (5/6)^n     where n is the number of trials

The probability of requiring at least two trials to observe a 3 is the sum of the probabilities of observing a 3 on the third, fourth, fifth, etc. trials. This is equivalent to the sum of an infinite geometric series. The sum can be calculated using the formula

S = a/(1 - r)        where S is the sum, a is the first term, and r is the common ratio.

In this case, a = 1 - (5/6)^2 and r = 5/6. So, the sum is S = (1 - (5/6)^2)/(1 - 5/6) = 6/36 = 1/6. Therefore, p(b) = 1/6.

(c) The probability of observing a 3 on the first trial or at least two trials is the sum of p(a) and p(b). So, p(a or b) = p(a) + p(b) = 1/6 + 1/6 = 1/3.

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