Answer:
Step-by-step explanation:
To find out how much shorter the pencil was after the student completed the project, we need to subtract the final length from the initial length:
7 1/2 - 5 7/8
To subtract these two mixed numbers, we need to convert them to a common denominator. The smallest common multiple of 2 and 8 (the denominators of the fractions) is 8, so we can rewrite the numbers as:
15/2 - 47/8
Now we can find a common denominator of 8:
(15/2) * (4/4) - (47/8) = 30/8 - 47/8
Subtracting the numerators, we get:
-17/8
Therefore, the pencil was 17/8 inches shorter after the student completed the project than when the student started the project. We can simplify this fraction to a mixed number:
-2 1/8
So the answer is C) 2 3/8 inches.
Solve three and one-half times one and two-thirds equals blank
Three and one-half times one and two-thirds is equal to 35/6 or 5 5/6.
What is the simplified form of three and one-half times one and two-thirds?Given the statement in the question; "three and one-half times one and two-thirds"
First, we can be written as a fraction:
Three and one-half → 3 1/2 = 7/2
One and two-thirds → 1 2/3 = 5/3
So, we have:
(7/2) × (5/3)
To multiply these fractions, we multiply the numerators together and the denominators together:
( 7 × 5 ) / ( 2 × 6 )
35/6
5 5/6
Therefore, the simplified form is 35/6 or 5 5/6.
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5. The function f(x) is defined by f(x)= [(x²-6,x<0) and (10-x,x>or equal to 0) state the domain and the range
PLEASE HELP ME ASAP!!!
The line should cross over the x-axis at the point (0.5,). The line should cross over the y-axis at the point (0,2). To show this on the graph, like the question is asking, you can place one point at (0,2) and another point at (1,-2).
Find the length of a rectangular lot with a perimeter of 110 m if the length is 5 m more than the width.
The length of the rectangular lot is 30m
How to determine the valueThe formula for calculating the perimeter of a rectangle is expressed as;
P =2(l + w)
Such that the parameters are;
P is the perimeter of the rectangle.l is the length of the rectangle.w is the width of the rectangle.From the information given, we have that;
Length = 5 + w
Now, substitute the values
110 = 2( 5 + w+ w)
collect the like terms
110 = 2(5 + 2w)
expand the bracket
110 = 10 + 4w
4w = 100
Divide both sides by the coefficient of w
w = 25
Then, the length = 30m
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Glorious Gadgets is a retailer of astronomy equipment. They purchase equipment from a supplier and then
sell it to customers in their store. The function C(x) = 1.5x +25500x¹+17000 models their total
inventory costs (in dollars) as a function of a the lot size for each of their orders from the supplier. The
inventory costs include such things as purchasing, processing, shipping, and storing the equipment.
What lot size should Glorious Gadgets order to minimize their total inventory costs? (NOTE: your answer
must be the whole number that corresponds to the lowest cost.)
What is their minimum total inventory cost? $
Glorious Gadgets should order a lot size of 1 to minimize their total inventory costs, and the minimum total inventory cost is $42,501.50.
What is rate?In mathematics, rate is commonly used in the context of describing how one quantity changes with respect to another quantity over time. For example, the rate of change of an object's position over time is its velocity, which is expressed in units of distance per unit of time (such as meters per second). Similarly, the rate of growth or decline of a population over time is often expressed as a percentage, such as an annual growth rate of 2% or a population decline rate of 1.5% per year.
According to the given information:To find the lot size that minimizes the total inventory cost, we need to find the value of "x" that minimizes the function C(x) = 1.5x + 25500x¹ + 17000.
To do this, we can take the derivative of C(x) with respect to x, set it equal to zero, and solve for "x":
C'(x) = 1.5 + 25500 + 0 = 25501.5
25501.5 = 0
x = -25500/1.5 = -17000
Since a negative value of "x" doesn't make sense in the context of the problem, we can conclude that there is no minimum value for C(x). Instead, the function C(x) increases as "x" increases, so Glorious Gadgets should order as small a lot size as possible to minimize their inventory costs.
To find the minimum total inventory cost, we can evaluate the function C(x) at a small lot size, such as x = 1:
C(1) = 1.5(1) + 25500(1)¹ + 17000
C(1) = 1.5 + 25500 + 17000
C(1) = 42501.5
Therefore, Glorious Gadgets should order a lot size of 1 to minimize their total inventory costs, and the minimum total inventory cost is $42,501.50.
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Homework: 7.10 Surface Area
1. Use the net of the rectangular prism below to find its surface area.
10 m
Show all work.
6 m
5m
Side
10 m
Top
Front
Top
Front
Sm
Side
6 m
Therefore, the surface area of the rectangular prism is 280 square meters.
What is area?Area is a measure of the size of a two-dimensional region or surface, typically expressed in square units. It is the amount of space inside a closed shape or figure, and is calculated by multiplying the length and width of the shape or by using a formula specific to the shape. For example, the area of a rectangle can be found by multiplying its length and width, while the area of a circle can be found by multiplying pi (3.14) by the square of its radius. Area is commonly used in mathematics, science, engineering, and other fields to calculate quantities such as surface area, volume, and density.
Here,
The rectangular prism has 6 faces, and each face is a rectangle. To find the surface area, we need to find the area of each rectangle and add them up. The net shows that the dimensions of the prism are:
length = 10 m
width = 6 m
height = 5 m
The area of the top and bottom faces (which are congruent) is:
A = length × width
= 10 m × 6 m
= 60 m²
The area of the front and back faces (which are congruent) is:
A = length × height
= 10 m × 5 m
= 50 m²
The area of the two side faces (which are congruent) is:
A = width × height
= 6 m × 5 m
= 30 m²
To find the surface area, we add up the areas of all six faces:
Surface Area = 2 × (top and bottom faces) + 2 × (front and back faces) + 2 × (side faces)
Surface Area = 2(60 m²) + 2(50 m²) + 2(30 m²)
Surface Area = 120 m² + 100 m² + 60 m²
Surface Area = 280 m²
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You are playing a game where you draw a card from a standard deck, and you will win $5 if you draw a face
card, $14 if you draw an ace and lose $19 if you draw any other card. What is the expected gain from this
game?
Since there are 12 face cards (4 J, 4 Q, 4K) those represent 12/52. The nonface cards excluding aces (4 twos through 4 tens) represent 36/52.
The expected value is then:
(12/52)($3)+(36/52)(-$2)= -$36/52 or -$0.69 but rounding to one decimal would be -$0.70
1. The grid represents 1. What does the shaded region represent?
Be prepared to explain your reasoning.
So the shaded region represents 30% of the whole. Each individual square on the grid must represent 1/100 of the whole.
What is grid?A grid is a network of horizontal and vertical lines that intersect at regular intervals to form a pattern of squares, rectangles, or other shapes. It is commonly used as a tool for visualizing and organizing information, such as in maps, charts, graphs, and other types of diagrams. A grid can be simple or complex, depending on its purpose and the level of detail required. Simple grids may consist of only a few lines, while complex grids may have many lines and sub-divisions. In many cases, grids are labeled with coordinates or numbers to facilitate navigation or data analysis.
Here,
If the grid represents 1 in a grid of 100 squares, then each individual square on the grid must represent 1/100 of the whole.
The shaded region covers 30 squares out of the 100 squares in the grid. So the shaded region represents a fraction of the whole that can be calculated by dividing the number of shaded squares by the total number of squares:
30/100
This fraction can be simplified to:
3/10
Therefore, the shaded region represents three-tenths (3/10) of the whole. Alternatively, we can also express this as a percentage by multiplying the fraction by 100:
(3/10) x 100 = 30%
So the shaded region represents 30% of the whole.
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PLEASE HELP I HAVE TO TURN THIS IN TODAY
A fair number cube is labeled with the numbers 1 – 6. After rolling the cube 100 times, which would be most likely to occur? Responses The number 6 would never be rolled. The number 6 would never be rolled. All the numbers rolled would be odd. All the numbers rolled would be odd. An even number would be rolled 50 times. An even number would be rolled 50 times. The number 7 would be rolled at least once. The number 7 would be rolled at least once.
The outcome that is most likely to occur after rolling the cube 100 times is that an even number would be rolled around 50 times.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
A fair number cube labeled with the numbers 1 - 6 is equally likely to land on any of its faces when rolled.
Therefore, we can use probability to determine which outcome is most likely to occur after rolling the cube 100 times.
"The number 6 would never be rolled." This outcome is highly unlikely, but not impossible.
If the cube is fair, each number has an equal probability of being rolled, so the probability of rolling a 6 on any given roll is 1/6.
The probability of rolling a non-6 on any given roll is 5/6. The probability of not rolling a 6 in 100 rolls is (5/6)¹⁰⁰, which is approximately 0.0001. So, while it is very unlikely that the number 6 would never be rolled, it is not impossible.
"All the numbers rolled would be odd." This outcome is even less likely than the first one.
The odd numbers on the cube are 1, 3, and 5, each with a probability of 1/2.
The probability of rolling an odd number on any given roll is 1/2.
The probability of rolling an odd number on 100 rolls is (1/2)¹⁰⁰, which is an extremely small number, approaching zero.
"An even number would be rolled 50 times." This outcome is more likely than the first two.
The even numbers on the cube are 2, 4, and 6, each with a probability of 1/2.
The probability of rolling an even number on any given roll is 1/2.
If we assume that each roll is independent of the others, the probability of rolling an even number exactly 50 times in 100 rolls can be calculated using the binomial distribution and is approximately 0.0796.
This means that this outcome is not very likely, but still possible.
"The number 7 would be rolled at least once."
This outcome is impossible since the cube is only labeled with the numbers 1 through 6.
Therefore, the number 7 cannot be rolled.
Hence, Based on the above analysis, the outcome that is most likely to occur after rolling the cube 100 times is that an even number would be rolled around 50 times.
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Solve for the variable. Determine if there is one solution, infinitely many solutions, or no solution.
-6x – 12 = 5(-2x – 3)
Answer:
Step-by-step explanation:
Solve for the variable. Determine if there is one solution, infinitely many solutions, or no solution.
-6x – 12 = 5(-2x – 3)
Let's start by simplifying the right side of the equation:
5(-2x – 3) = -10x - 15
Substituting this back into the original equation, we get:
-6x - 12 = -10x - 15
Next, we can simplify this equation by adding 10x to both sides:
4x - 12 = -15
Finally, we can add 12 to both sides:
4x = -3
Dividing both sides by 4 gives us:
x = -3/4
Therefore, there is one solution, and x equals -3/4.
A painting measures 1 meter by 2 meters. A frame shop charges $2.17 per meter for a wooden frame. How much would it cost to buy a frame for the painting?
It would cost $13.02 to buy a Rectangle type frame for the painting.
What exactly is a rectangle?
A rectangle is a two-dimensional geometric shape that is characterized by having four sides and four right angles. It is a type of quadrilateral.
The opposite sides of a rectangle are parallel and equal in length, and the adjacent sides are perpendicular to each other. This means that the opposite sides of a rectangle have the same distance apart throughout their entire length. The Perimeter is given by 2(L+B), where L and B are length and breadth respectively.
Now,
The painting measures 1 meter by 2 meters, so the perimeter of the painting (the total length of its sides) is:
P = 2 x (1 + 2) = 2 x 3 = 6 meters
To frame the painting, we need to buy a wooden frame with a perimeter of 6 meters. The frame shop charges $2.17 per meter for the wooden frame, so the cost of the frame would be:
Cost = 6 x $2.17 = $13.02
Therefore, it would cost $13.02 to buy a frame for the painting.
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what did I do wrong???????
Answer:
Step-by-step explanation:
Austin is designing a new board game, and is trying to figure out all the possible outcomes. How many different possible outcomes are there if he flips a coin, spins a spinner with 5 equal-sized sections labeled Monday, Tuesday, Wednesday, Thursday, Friday, and spins a spinner with four equal-sized sections labeled Red, Green, Blue, Orange?
Jerry had the following transaction on his credit card statement on April 1st is unpaid balance $5942.00 the on April 10th a purchase was made $124.00 on April 22 a payment was made $1000.00 on April 28 a purchase was made $628.00
Answer:
Starting unpaid balance = $5942.00
On April 1st: Starting unpaid balance = $5942.00
On April 10th: $124.00 purchase is made, so the new balance becomes $5942.00 + $124.00 = $6066.00
On April 22nd: $1000.00 payment is made, so the new balance becomes $6066.00 - $1000.00 = $5066.00
On April 28th: $628.00 purchase is made, so the new balance becomes $5066.00 + $628.00 = $5694.00
Therefore, the ending balance on Jerry's credit card statement at the end of April is $5694.00.
Step-by-step explanation:
PLEASE HELP!!!
How do you write a mnumatic expression? (Yes the M is supposed to go there)
Mnemonic expressions can be useful for remembering mathematical concepts, formulas, and equations.
Some examples of mnemonic expressions in mathematics:PEMDAS: This acronym stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It is used to remember the order of operations in mathematical calculations.
SOH-CAH-TOA: This acronym is used to remember the trigonometric ratios for sine, cosine, and tangent. SOH stands for Sine = Opposite/Hypotenuse, CAH stands for Cosine = Adjacent/Hypotenuse, and TOA stands for Tangent = Opposite/Adjacent.
FOIL: This acronym is used to remember the process of multiplying two binomials. It stands for First, Outer, Inner, Last, and represents the order in which you multiply the terms of the two binomials.
Please Excuse My Dear Aunt Sally: This sentence is used to remember the order of operations in mathematical calculations. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
My Dear Aunt Sally Loves Ice Cream: This sentence is used to remember the signs of trigonometric functions in each quadrant of the unit circle. It stands for positive for cosine and ice cream, negative for sine and loves, and alternating for tangent.
These are just a few examples of mnemonic expressions in mathematics, and there are many more that can be used to help remember mathematical concepts and formulas.
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Forces of 500 kg and 875 kg act on an object. The angle between the forces is 100 deg Find the resultant, giving the angle it makes with the smaller force
Answer:
To find the resultant of two forces, you can use the following formula:
R^2 = F1^2 + F2^2 + 2F1F2cosθ
Where R is the resultant force, F1 and F2 are the magnitudes of the two forces, θ is the angle between the two forces, and cosθ is the cosine of the angle.
Plugging in the given values, we get:
R^2 = (500 kg)^2 + (875 kg)^2 + 2(500 kg)(875 kg)cos(100 deg)
R^2 = 250000 + 765625 + 767321.98
R^2 = 1788946.98
R = sqrt(1788946.98) ≈ 1337.3 kg
So the magnitude of the resultant force is approximately 1337.3 kg.
To find the angle that the resultant force makes with the smaller force, we can use the following formula:
tanθ = (F2sinθ)/(F1 + F2cosθ)
Where θ is the angle between the resultant force and the smaller force.
Plugging in the given values, we get:
tanθ = (875 kg sin(100 deg))/(500 kg + 875 kg cos(100 deg))
tanθ = 0.768
θ = tan^-1(0.768) ≈ 36.2 deg
So the angle that the resultant force makes with the smaller force is approximately 36.2 degrees.
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5. Lisa's age plus Sean's age is 17. Sean is 11 years old. How old is Lisa?
Answer:
Lisa is 6 years old.Step-by-step explanation:
We are given that Lisa's age plus Sean's age is 17 years and Sean is 11 years old .
We need to find the age of Lisa.
Let's assume age of Lisa be x years.
A/q Lisa's age plus Sean's age is 17
[tex] \longrightarrow \: x + Sean's \: age = 17 \\ \\ \longrightarrow \: x + 11 = 17 \\ \\ \longrightarrow \: x = 17 - 11 \\ \\ \longrightarrow \: x = 6 \\ \\ [/tex]
Hence, Lisa is 6 years old.
How can I show my son a simple way of expressing 5/8 to a decimal fraction
Answer:
0.625
Step-by-step explanation:
you divide 5 by 8
Jason conducted a survey amongst his friends to determine the amount of hours they watch TV each week. Here is the table that Jason found.
What statement is NOT true about the frequency chart?
Answer:
70% watched less than an hour.
Step-by-step explanation:
Less than 1 hour -- 3 people
Number of people 10
3/10 -- 30% of people watch less than 1 hour.
Find X and Y if X + Yi = 3Y - 2Xi + 4i
For the given equation, the values of X and Y are X=4 and Y=-2, respectively.
What is equation?
In mathematics, an equation is a statement that asserts the equality of two expressions. It consists of two expressions separated by an equal sign (=).
According to given information:To find the values of X and Y, we need to separate the real and imaginary parts of the complex number on the right-hand side of the equation.
We have:
X + Yi = 3Y - 2Xi + 4i
Separating the real and imaginary parts, we get:
X = -2Y + 0
Y = 3Y + 4
Simplifying the second equation, we have:
-2Y = 4
Dividing by -2, we get:
Y = -2
Substituting this value of Y into the first equation, we have:
X = -2(-2) = 4
Therefore, the values of X and Y are X=4 and Y=-2, respectively.
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a grocery store sells packages for fresh tortillas . the table shows the relationship between the number of packages and the number of packages and the number of tortillas . how many are in 2 packages
The Tortillas in 2 Packages from Given Data is 14
Calculating Tortillas in 2 Packages from Given DataTo determine how many tortillas are in 2 packages, we can use the given data to find the relationship between the number of packages and the number of tortillas.
From the table, we can see that there is a linear relationship between the two variables, with each package containing 7 tortillas.
Therefore, we can calculate the number of tortillas in 2 packages by multiplying 2 by 7,
So, we have
Packages = 2 * 7
Packages = 14
The answer of 14 tortillas in 2 packages.
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Complete question
A grocery store sells packages of fresh tortillas. The table shows the relationship between the number of packages and the number of tortillas. How many tortillas are in 2 packages?
Number of Packages Number of Tortillas
5 35
7 49
15 105
Classify the polynomial by degree and number of terms:
y = 2x5-18x²
Answer: Y=10
Step-by-step explanation: please label me brainiest if I'm right
Multiply write your answer in SIMPLEST form PLS FAST
According to your query I am defining you multiply in simple manner.
What is the term multiply?The term multiply refers to the mathematical operation of repeatedly adding a number to itself a certain number of times. The result of this operation is known as the product. Multiplication is an essential arithmetic operation that is used in various fields, including science, engineering, and economics.
Multiplication is typically represented by the symbol "×" or "*", and the numbers being multiplied are called factors. For example, if we multiply 2 and 3, we get the product 6, which can be written as 2 × 3 or 2 * 3.
Multiplication has several important properties, including the commutative property, which states that the order of the factors does not affect the product. For example, 2 × 3 is equal to 3 × 2.
Another important property of multiplication is the distributive property, which states that the product of a number and the sum of two or more numbers is equal to the sum of the products of the number and each of the individual numbers. For example, 2 × (3 + 4) is equal to (2 × 3) + (2 × 4), which simplifies to 14.
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According to your query when you multiply in simple manner and it simplifies to 7/5.
What is the term multiply?The term multiply refers to the mathematical operation of repeatedly adding a number to itself a certain number of times. The result of this operation is known as the product. Multiplication is an essential arithmetic operation that is used in various fields, including science, engineering, and economics.
Multiplication is typically represented by the symbol "×" or "*", and the numbers being multiplied are called factors. For example, if we multiply 2 and 3, we get the product 6, which can be written as 2 × 3 or 2 * 3.
For the given question:
11/7 × 49/5
= 539/35
= 7/5
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The complete question is as follows:
Question 2b: Which equations are parallel to the given line? y = 6x -3 Select ALL that apply A. y = -6x -3 B. y = 6x -4 C. 12x - 2y = -4 D. 6y = -x + 5 E. 12x + 2y = -4
Answer: B. y= 6x -4 and C. 12x -2y=-4
Step-by-step explanation:
I got it wrong on nearpod. but luckily nearpod shows you the correct answer after submitting the answer choices. So these are the actual answers.
The prism below is made of cubes which measure 2 of an inch on one side. What is the volume? OA. 6 cubic in OB. OC. 312 7 cubic in cubic in OD. 12 cubic in Note: Figure is not drawn to scale.
The required volume of the given cuboid is (B) 3/2 inches³.
What is a cuboid?A cuboid is a solid shape or a three-dimensional shape in geometry.
A cuboid is a convex polyhedron that has six rectangular faces, eight vertices, and twelve edges.
Another name for a cuboid is a rectangular prism.
A cube is a cuboid with six square faces.
For instance, furniture, books, and televisions all have rectangular frames. It is a cuboid shape.
On the other hand, the 3D shape known as a cube is represented by items like ice, dice, and Rubik's cube. In actuality, a cube is a particular kind of cuboid with square, identical sides.
So, the volume of cuboid:
V= l*w*h
V = 1*1.5*1
V = 1.5
V = 1/5 = 3/2
Therefore, the required volume of the given cuboid is (B) 3/2 inches³.
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Use the definition of a logarithm to solve the equation. ln ( − 5 z ) = ln ( z ^2 − 7 z )
Answer:
Step-by-step explanation:
From the property of logarithm the input cannot be negative.
Therefore Z should be negative
As from the given question both sides have the (ln)
operator thus we can remove it.
-5Z=z^2-7Z
2Z=Z^2
Z^2=2Z
Z=2
advise zanele on whether or not to take a gap year. questionnaire suggested for zanele. 1. what is your name? 2. why do you think that it is wise to take a gap year after school? 3. what are the benefits of taking a gap year ? 4. what are the reason for delaying a gap year than first pursuing a university degree? 5. what is the best way to spend a gap year? after working through the suggested questions zanele realises that the questionnaire is not suitable. list two reasons why you think that the given questionnaire is not suitable.
If Zanele realized that the questiοnnaire is nοt suitable, it cοuld be fοr variοus reasοns, such as:
The questiοns may nοt be relevant tο her specific situatiοn οr gοals.The questiοns may nοt prοvide enοugh infοrmatiοn οr οptiοns tο help her make an infοrmed decisiοn.What are the impοrts οf the questiοns οn the questiοnnaire?What is yοur name?
Name may nοt be needed tο prοvide advice οn taking a gap year.
Why dο yοu think that it is wise tο take a gap year after schοοl?
Taking a gap year after schοοl can prοvide students with a break frοm academic studies, help them gain life experience, and give them time tο explοre their interests befοre cοmmitting tο a university degree.
What are the benefits οf taking a gap year?The benefits οf taking a gap year include persοnal and academic grοwth, increased indeοendence, cultural exοοsure, and the οοοοrtunity tο gain valuable wοrk οr vοlunteer exοerience.
What are the reasοns fοr delaying a gap year than first pursuing a university degree?Sοme students chοοse tο delay taking a gap year tο οursue a university degree first sο that they can stay οn track with their academic and career gοals. Others may want tο save mοney οr avοid interruοting their academic mοmentum.
What is the best way tο spend a gap year?The best way tο spend a gap year deοends οn yοur interests and gοals. Sοme pοpular gap year activities include traveling, vοlunteering, interning, wοrking, learning a new language οr skill, οr pursuing a passiοn prοject.
If Zanele realized that the questiοnnaire is nοt suitable, it cοuld be fοr variοus reasοns, such as:
The questiοns may nοt be relevant tο her specific situatiοn οr gοals.
The questiοns may nοt prοvide enοugh infοrmatiοn οr οptiοns tο help her make an infοrmed decisiοn.
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Can someone please help me
Answer: [1] G,D,L,K [2] HF and AB [3] X-Y Plane [4] CE and LA
[5] A,L,B
Step-by-step explanation:
For each of the following, list the sample space and tell whether you think the events are equally likely: b) A family has 3 children; record each child's sex in order of birth c) Toss four coins; record the number of tails d) Toss a coin 10 times; record the length of the longest run of heads
Solution for b: The events "all boys," "all girls," "2 boys and 1 girl," and "2 girls and 1 boy" are equally likely because each has probability 1/8. Solution for c: The events "0 tails," "1 tail," "2 tails," "3 tails," and "4 tails" are equally likely because each has probability 1/16. Solution for d: the events are not equally likely.
b) For a family having 3 children, the sample space is {BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}, where B represents a boy and G represents a girl. Each outcome in the sample space has probability 1/8. The events "all boys," "all girls," "2 boys and 1 girl," and "2 girls and 1 boy" are equally likely because each has probability 1/8.
c) For tossing four coins, the sample space is {TTTT, TTTH, TTTT, TTHH, TTHH, THHH, THHT, THHH, HHHH, HHTH, HHTT, HHHT, HHTH, HTHH, HTTH, HTTT}, where T represents tails and H represents heads. Each outcome in the sample space has probability 1/16. The events "0 tails," "1 tail," "2 tails," "3 tails," and "4 tails" are equally likely because each has probability 1/16.
d) For tossing a coin 10 times, the sample space is all possible sequences of 10 H's and T's. There are 2^10 = 1024 outcomes in the sample space, each with probability 1/1024. The events "longest run of heads is 1," "longest run of heads is 2," "longest run of heads is 3," "longest run of heads is 4," "longest run of heads is 5," and "longest run of heads is 6 or more" are not equally likely, because the probability of getting a long run of heads is much lower than the probability of getting a short run of heads. For example, the probability of getting a run of 6 heads in a row is (1/2)^6 = 1/64, while the probability of getting a run of 1 head is 1/2. Therefore, the events are not equally likely.
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