The four roots form the vertices of a square in the complex plane. The side length of the square is √2. The vertices of the square complex number are: A = (-√2, -√2), B = (0, -2√2), C = (√2, -√2), D = (0, 0). The principal root is the one with the smallest argument, which is (-√2 - √2 i).
To find the fourth root of the complex number (-8-8√3), we can use the following steps:
We start by finding the modulus and argument of the number:
|(-8-8√3)| = 16
arg(-8-8√3) = arctan(-√3) = -π/3
Therefore, (-8-8√3) = 16(cos(-π/3) + i sin(-π/3)).
To find the fourth root, we take the square root twice. We can use De Moivre's theorem to simplify the calculation:
(-8-8√3)^(1/4) = (16(cos(-π/3) + i sin(-π/3)))^(1/4)
= 2(cos(-π/12) + i sin(-π/12))
= 2(cos(11π/12) + i sin(11π/12))
= 2(-√2/2 - √2/2 i)
We have expressed the roots in rectangular coordinates as (-√2, -√2), (0, -2√2), (√2, -√2), and (0, 0).
The four roots form the vertices of a square in the complex plane. The side length of the square is |(-√2) - (0)| = √2.
The vertices of the square are:
A = (-√2, -√2)
B = (0, -2√2)
C = (√2, -√2)
D = (0, 0)
The principal root is the one with the smallest argument, which is (-√2 - √2 i). This is the root corresponding to the first quadrant of the complex plane.
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A committee must be formed with 5 teachers and 4 students. If there are 11 teachers to choose from, and 14 students, how many different ways could the committee be made?
Answer:
there are 462,462 different ways the committee can be formed.
Step-by-step explanation:
To form the committee, we need to choose 5 teachers out of 11 and 4 students out of 14. We can use the combination formula to find the number of ways:
Number of ways to choose 5 teachers out of 11:
C(11, 5) = 11! / (5! * 6!) = 462
Number of ways to choose 4 students out of 14:
C(14, 4) = 14! / (4! * 10!) = 1001
The total number of ways to form the committee is the product of these two combinations:
462 * 1001 = 462462
Therefore, there are 462,462 different ways the committee can be formed.
A credit card has a balance of $3,400. The APR is 27% and the minimum payment is 3% of the balance. You will pay the minimum balance this month. If you do not use the card again then how much should the balance be next month?
The minimum payment for this month would be:
minimum payment = 3% of $3,400 = 0.03 x $3,400 = $102
The interest charged on the remaining balance after the minimum payment is made can be calculated as:
interest = APR/12 x remaining balance
where APR is the annual percentage rate and the division by 12 is to convert it to a monthly rate.
So for this month, the interest charged would be:
interest = 0.27/12 x $3,298 = $74.32
where $3,298 is the remaining balance after the minimum payment is made.
Therefore, the balance next month would be:
balance = previous balance - payment + interest
balance = $3,400 - $102 + $74.32 = $3,372.32
So the balance next month should be $3,372.32 if you only make the minimum payment and do not use the card again.
||
Initial Knowledge Check
Simplify.
(4w)²
Write your answer without parentheses.
01
6
X
Ś
The expression is 16w².
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
(4w)²
This can be expressed as,
= 4² x w²
[ 4 x 4 = 16 ]
= 16 x w²
= 16w²
Thus,
The expression is 16w².
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PLeasee ANSWER will mark brainliest
Answer:
The insect population increased at a rate of 20% each month.
Step-by-step explanation:
i looked it up on gogle and it popped up on brainly asked by someone else and already answered
Sam went tpbschool at 8 15. he was in homeroom for 20 minutes. math for 35 minutes and reading for 35 minutes and then had a 15 minute break. how long was sam at school?
Sam spent 1 hour and 55 minutes at school.
What is addition?Combining objects and counting them as one big group is done through addition. In arithmetic, addition is the process of adding two or more integers together. Addends are the numbers that are added, and the sum refers to the outcome of the operation.
Given:
Sam went to school at 8:15.
He was in homeroom for 20 minutes, math for 35 minutes, and reading for 35 minutes and then had a 15-minute break.
The total time is,
= 20 + 35 + 35 + 25
= 115 minutes.
= 1 hour and 55 minutes.
Therefore, Sam spends 1 hour and 55 minutes at school.
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In a lottery game, a player picks 6 numbers from 1 to 48. if 4 of those 6 numbers match those drawn, the player wins third prize. let's walk through the steps to determine the probability of winning third prize.In how many ways can 4 winning numbers be chosen from the possible 6 numbers?In how many ways can 2 non-winning numbers be chosen from the pool of all non-winning numbers? The number of favorable outcomes would be to multiply the above two answers together, since we want 4 winning numbers and 2 non-winning numbers. What is the number of favorable outcomes? In how many ways can you pick any 6 numbers from the pool of 50 numbers? This is your total outcomes. What is the probability of winning third prize?
The probability of winning third prize in the lottery game is 0.0011 or about 0.11%.
In the given lottery game, the player picks 6 numbers from 1 to 48.
The number of ways 4 winning numbers can be chosen from the possible 6 numbers is calculated using the combination formula: C(6,4) = 15.The number of ways 2 non-winning numbers can be chosen from the pool of 42 non-winning numbers (total numbers - possible winning numbers) is calculated using the combination formula: C(42,2) = 861.The number of favorable outcomes is the product of the above two answers: 15 x 861 = 12,915.The total number of ways you can pick any 6 numbers from the pool of 48 numbers is calculated using the combination formula: C(48,6) = 12,271,512.The probability of winning third prize is the number of favorable outcomes divided by the total number of outcomes: 12,915 / 12,271,512 ≈ 0.0011 or about 0.11%. Therefore, the probability of winning third prize in this lottery game is relatively low.Learn more about probability:
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find the inverse of each function f(x) = x-3/2
Find the value of the constant that makes the following function continuous on (−∞,∞).
The piecewise function is continuous only if a = 2.
Which must be the value of the constant a?The function will be continuous if both parts of the function have the same value on the "jump" between the two parts.
The first part evaluated in x = 1 is:
f(x) = (4x^3 - 4x)/(x - 1)
We can rewrite the numerator as:
(4x^3 - 4x) = 4x*(x^2 - 1) = 4x*(x + 1)*(x - 1)
Then we can rewrite the function as:
f(x) = 4x*(x + 1)
Evaluating in x = 1 we will get:
f(1) = 4*1*(1 + 1) = 8
Then the other piece evaluated in x = 1 also must be equal to 8, we will get:
f(x) = 4x^2 + 2x + a
f(1) = 4*1^2 + 2*1 + a = 8
= 4 + 2 + a = 8
a = 8 - 4 - 2 = 2
The value of a must be 2.
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There are 12 inches in 1 foot. Which of the following conversion factors express this relationship correctly? Select all that apply Check all that apply. a.12 in /1 ft b. 1 in x 12 ft c. 1 ft /12 in d.1/ 1 in x 12 ft
The relationship exhibiting conversion factor correctly is the option c. 1 ft /12 in, where ft is foot and in is inches.
As per the mentioned information, the relationship will be expressed as -
12 inches = 1 foot
So, now we will use unitary method to find 1 inches, which will be the conversion factor.
Now, the formula for conversion factor (that is one unit of inches) will be = number of unit in foot/number of unit in inches
Conversion factor = 1/12
Based on this, the correct option indicating conversion factor is c. 1 ft /12 in.
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A right, isosceles triangle has hypotenuse length 50cm. How long are the two legs?
The length of the two legs is 25√2 cm
How to determine the length of the two legs?From the question, we have the following parameters that can be used in our computation:
A right, isosceles triangle has hypotenuse length 50cm
Represen the length of each leg with x
Using the above definition as a guide, we have the following:
x = Hypotenuse/√2
So, we have
x = 50/√2
This gives
x = 25√2
HEnce, the legs are 25√2 cm each
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A plane is 107 mi north and 172 mi east of an airport. Find x, the angle the pilot should turn in order to fly directly to the airport. Round your answer to the
nearest tenth of a degree.
The pilot should turn approximately 50.7° to fly directly to the airport.
How to find the angle the pilot should turnThe problem can be solved using trigonometry.
Let x be the angle the pilot should turn in order to fly directly to the airport.
Then, we have:
tan(x) = 107 / 172
x = arc tan^-1(107 / 172)
x = 50.7 degrees
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If b is a positive real number and mand n are positive integers, then
fm/n
√bm = (vb)m
=
OA. True
OB. False
The given statement for b is a positive real number and m and n are positive integers, is true.
What is an integer?An integer is a number with no decimal or fractional part, and it includes negative and positive numbers, including zero. A few examples of integers are: -5, 0, 2, -8...
Given that, b is a real number and m and n are positive integers.
According to the above information, the calculation is as follows:
[tex]b^\frac{m}{n} = \sqrt[n]{b^m} = (\sqrt[n]{b} )^m[/tex]
Hence, the statement is true.
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Aaron is trying to decide which job would allow him to earn the most money after a few years.
Job A agrees to pay him $35,000 in his first year and they promise to give him a 2% raise each year.
Job B can be modeled by the equation:
B(t)=30,000(1.03)t
Which job offers the largest initial salary?
Which job offers the fastest rate of increase?
Calculate Aaron’s salary after 20 years for each job offer.
Which job offer should Aaron accept and why?
So, Lan's total monthly insurance expense is $221.67.
What is time period?A time period is a duration of time characterized by specific events, trends, or conditions. Time periods can be defined by various factors such as historical, cultural, or scientific significance. Examples of time periods include the Renaissance, the Industrial Revolution, the Bronze Age, and the Jurassic period in Earth's geological history. Time periods can vary in length from a few years to millions of years, depending on the context in which they are being discussed.
Given by the question: -
First, let's prorate the semiannual premium for automobile insurance.
Since semiannual means twice a year, we need to divide $450 by 2 to get the premium for a single 6-month period:
$450 ÷ 2 = $225
To find the monthly premium, we need to divide this by 6 (since there are 6 months in a 6-month period):
$225 ÷ 6 = $37.50
So, the monthly premium for automobile insurance is $37.50.
Next, we don't need to prorate the monthly premium for health insurance since it is already a monthly expense. Therefore, the monthly premium for health insurance is simply $155.
Finally, to prorate the annual premium for life insurance, we need to divide it by 12 to get the premium for a single month:
$350 ÷ 12 = $29.17
So, the monthly premium for life insurance is $29.17.
Therefore, the total monthly expense for these three types of insurance is: $37.50 + $155 + $29.17 = $221.67
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The length of a rectangular poster is 4 more inches than three times its width. The area of the poster is 340 square inches. Solve for the dimensions (length and width) of the poster.
31 in by 31/4 in is the required dimension of the poster
Determining the area of a rectangleThe formula for calculating the area of a rectangle is expressed as:
A = lw
If the length of a rectangular poster is 4 more inches than three times its width, then;
l = 3w - 4
Substitute
340 = w(3w - 4)
340 = 3w^2 - 4w
3w^2 - 4w - 340 =0
On factorizing, the measure of the width of the poster is 34/3 in
l = 3w - 4
l = 3(34/3) - 4
l = 31in
Hence the dimension of the given poster is 31 in by 31/4 in
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Solve for x.
x2 - 2x - 24 = 0
[tex] \bf \: {x}^{2} - 2x - 24 = 0[/tex]
[tex] \bf \: {x}^{2} - 6x + 4x- 24 = 0[/tex]
[tex] \bf \: {x}^{}( x- 6) + 4(x- 6) = 0[/tex]
[tex] \bf \: {}^{}( x + 4)(x- 6) = 0[/tex]
[tex]x = - 4 \: \: or \: \: 6[/tex]
Referring to the Fig. in Question #23, find the cosine of angle S. Reduce the answer to the lowest terms.
The cosine of angle S is 4/5.
Describe Cosine?In mathematics, cosine is a trigonometric function that relates the angles of a right-angled triangle to the ratio of the length of its adjacent side to its hypotenuse.
More formally, cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse of a right-angled triangle containing that angle.
The cosine of an angle theta, denoted as cos(theta), is given by the formula:
cos(theta) = adjacent side / hypotenuse
The cosine function has a range of values between -1 and 1, and is periodic with a period of 2π radians or 360 degrees.
The ratio of the neighboring side to the hypotenuse is known as the cosine of an acute angle in a right triangle.
In triangle RST, the hypotenuse is RS = 10 and the adjacent side to angle S is ST = 8.
Therefore, the cosine of angle S is:
cos(S) = adjacent/hypotenuse = ST/RS = 8/10
To simplify this fraction to lowest terms, we can divide both the numerator and denominator by their greatest common factor, which is 2.
8/10 = (8/2)/(10/2) = 4/5
So, the cosine of angle S is 4/5.
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A programmer erroneously wrote an expression as: 0 < x < 10. Rewrite the expression using logical AND. Use parentheses. (x > 0)
A programmer erroneously wrote an expression as 0 < x < 10. This expression 0 < x < 10 can be rewritten using logical AND as (0 < x) AND (x < 10).
The expression 0 < x < 10 is commonly used to indicate that x is a value between 0 and 10, but it is not correct in programming. In programming, logical operations like "less than" and "greater than" are usually represented by < and > symbols, respectively.
To properly express the condition that x is between 0 and 10, we need to use the logical operator AND, which allows us to combine multiple conditions.
The logical AND operator in programming is typically represented by the symbol && or the word "AND". It evaluates to true if and only if both of the conditions on either side of the operator are true. If either condition is false, the whole expression is false.
To rewrite the expression 0 < x < 10 using logical AND, we first separate the conditions into two separate expressions:
x > 0 AND x < 10
which can be written as,
(0 < x) AND (x < 10).
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Describe the functions on the sections of the graph labeled on the coordinate plane below. Explanation too
Sections A, B, C, and D represent parabolic, linear with a positive slope, linear with a negative slope, and parabolic function, respectively.
What is a function?A function is a statement, conception, or regulation that establishes an association between two parameters. Functions may be found throughout mathematics and are essential for the development of significant links.
The straight line represents the linear function.
As 'x' increases, the value of 'y' also increases, then the slope of the line function is positive.As 'x' increases, the value of 'y' also decreases, then the slope of the line function is negative.The curve represents the parabolic function.
Sections A, B, C, and D represent parabolic, linear with a positive slope, linear with a negative slope, and parabolic function, respectively.
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Question 15 of 15 Select the equation and solution for this situation. 14 less than a number is 22. O A. x+ 14 = 22 x = 8 OB. X-14 = 22 X = 36 O C. X-14 = 22 X = 8 O D. x + 14 = 22 x = 36 QUICK DUE TODAY
The solution to the equation x - 14 = 22 is x = 36
What is an equation?An equation is an expression that shows the relationship between numbers and variables using mathematical operators like addition, subtraction, multiplication and division. Equations can be linear, quadratic.
Let x represent the number. 14 less than a number is 22, hence:
x - 14 = 22
Add 14 to both sides of the equation:
x - 14 + 14 = 22 + 14
x = 36
The solution to the equation is x = 36
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Please explain your answer. Will mark Brainliest (question 9)
The initial investment of the function is 20 dollars.
The rate growth in percentage is 4%.
The investment after 10 years is 29.6 dollars.
How to solve function?The function [tex]y=20(1.04)^{t}[/tex] represents the value y of a saving account after t years.
Therefore, the initial investment of the function is 20 dollars.
The rate growth in percentage can be calculated as follows;
rate growth in percent = 0.04 × 100
rate growth in percent = 4 %
Let's find the value of the investment after 10 years.
Therefore,
[tex]y=20(1.04)^{t}[/tex]
t = 10
[tex]y=20(1.04)^{10}[/tex]
[tex]y=20(1.48024428492)[/tex]
y = 29.6048856984
Therefore, the investment after 10 years is as follows:
y = 29.6 dollars
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can someone help please
The degree of the polynomial is 5 and the maximum number of real zeroes in the polynomial is 5.
What are polynomials?
A polynomial is a mathematical equation that solely uses the operations addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Variables are sometimes known as indeterminate in mathematics.
f(x) = 6x⁵ - 3x⁴ + 5x² + 4
According to the fundamental theorem of algebra, the polynomial with n number of degrees will have n number of zeroes and no more, so the zeroes will be the same here 5.
Therefore, The polynomial has a degree of 5, and it can have a maximum of five real zeroes.
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Help
An art class is making a mural for their school which has a triangle drawn in the middle. The length of the bottom
of the triangle is x. Another side is 11 more than four times the length of the bottom of the triangle. The last side is 6 more
than the bottom of the triangle. Write and simplify an expression for the perimeter of the triangle.
What is the expression for the perimeter of the triangle?
The expression for the perimeter of the triangle 6x+17.
What is perimeter?Perimeter the sum of length of the sides used to made the given figure. A regular figure with n-sides has n equal sides in it, and they are the only parts of it(that means, nothing more than those equal lengthened n sides).
We have Given the length of the bottom of the triangle = x.
Since other side is 11 more than four times the length of the bottom of the triangle.
Also we have last side is 6 more than the bottom of the triangle.
Now, the sides of triangle;
x, 4x+11, x+6
Perimeter is the sum of lengths of all sides of figure.
x+ 4x+11+ x+6
=6x+17
Therefore, the perimeter expression will be 6x+17.
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24. Area of a Triangle = ½ Base x Height.
SAIL BOATS Tamara requests a
custom-made sail for her sailboat. The
base of her triangular sail is 2x + 1 and
the height is 4x + 6.
4x+6
2x+1
Find the area of the sail.
The area of sail is equivalent to (4x + 8x + 3) square units.
What is the area of a triangle?The area of a triangle is given as -
[tex]$A_{T} =\frac{1}{2} \times b \times h[/tex]
Given is a sail whose base is (2x + 1) units and height is (4x + 6) units.
We can write the area of sail as -
A = 1/2 x b x h
A = 1/2 x (2x + 1) x (4x + 6)
A = (2x + 1)(2x + 3)
A = 2x(2x + 3) + 1(2x + 3)
A = 3 + 2x + 6x + 4x²
A = 4x² + 8x + 3
Therefore, the area of sail is equivalent to (4x² + 8x + 3) square units.
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the national average for mathematics on a standardized test in 2011 was 516. suppose that the distribution of scores was approximately bell-shaped and that the standard deviation was approximately 43. round your answers to at least one decimal place as needed.
The rounded answer is that approximately 68% of scores were between 473 and 559.
To calculate this, we can use the formula for a normal distribution: P(μ−σ < X < μ+σ) = 68%, where μ is the mean and σ is the standard deviation.
For our problem, our mean is 516 and our standard deviation is 43. Therefore, our equation becomes P(473 < X < 559) = 68%. This means that 68% of scores were between 473 and 559, which is our answer.
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Use the given frequency distribution to find the (a) class width. (b) class midpoint of the first class. (c) class boundaries of the first class. Height (in inches) Class | Frequency, f 50-52 53-55 56-58 59-61 62-64 12 13 (a) 3 (c) 50-52 (a) 2 (c) 49.5-52.5 (a) 2 (b) 51.5 (c) 50-52 (a) 3 (b) 51 (c) 49.5-52.5 O A. (b) 51 OB, O C. (b) 51.5 ○ D. Click to select your answer.
The class width, class midpoint of the first class, and class boundaries of the first class of the given frequency distribution is 3, 51, and 49.5-52.5, respectively.
A frequency distribution is a table that shows how often different values or ranges of values occur in a set of data. In other words, it displays the frequency or count of each distinct value or range of values in a dataset.
To find the class width, we need to subtract the lower limit of the first class from the lower limit of the second class (since all the classes have the same width):
Class Width = Lower limit of second class - Lower limit of first class
= 53 - 50
= 3
Therefore, the class width is 3.
To find the class midpoint of the first class, we need to find the average of the lower limit and upper limit of the first class:
Class Midpoint = (Lower limit + Upper limit) / 2
= (50 + 52) / 2
= 51
Therefore, the class midpoint of the first class is 51.
To find the class boundaries of the first class, we first need to subtract the upper class limit for the first class from the lower class limit for the second class and divide it by two.
(Lower class limit of the second class - Upper class limit of the first class)/2
= (53 - 52)/2
= 0.5
Then, subtract the result to the lower limit of the first class and add to the upper class of the first class:
Class Boundaries = (Lower limit - 0.5) to (Upper limit + 0.5)
= (50 - 0.5) to (52 + 0.5)
= 49.5 to 52.5
Therefore, the class boundaries of the first class are 49.5-52.5.
Based on these calculations, the class width is 3, the class midpoint of the first class is 51, and the class boundaries of the first class is 49.5-52.5.
The problem seems incomplete, it must have been...
"Use the given frequency distribution to find the
(a) class width.
(b) class midpoint of the first class.
(c) class boundaries of the first class.
Height (in inches)
Class | Frequency, f
50-52 5
53-55 8
56-58 12
59-61 13
62-64 11
(a) 2 (b) 51.5 (c) 50-52(a) 2 (b) 51.5 (c) 49.5-52.5(a) 3 (b) 51 (c) 50-52(a) 3 (b) 51 (c) 49.5-52.5"Learn more about frequency distribution here: https://brainly.com/question/1094036
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Is y + 2x = 9 consistent or inconsistent
Yes, the given equation y + 2x = 9 is consistent.
What is slope intercept form of line?The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The equation y + 2x = 9 is a linear equation in two variables, x and y.
Now let us solve for y
y=-2x+9
We plot this equation graphically.
This equation represents a line with slope -2 and y-intercept 9.
So the graph of this equation is a straight line that passes through the point (0, 9) and has a slope of -2.
To determine if the equation y + 2x = 9 is consistent or inconsistent, we need to know if there exists at least one pair of values (x, y) that satisfy the equation.
We can see from the graph that there are infinitely many points on the line y + 2x = 9.
Hence, the given equation y + 2x = 9 is consistent.
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A piece of paper can be made into a cylinder in two ways: by joining the short sides together, or by joining the long sides together12. Which cylinder would hold more? How much more?
The solution is:
Bigger volume the cylinder wich radius is (L/2) and h = WHow much according to the relation L/W Let´s call L and W the dimensions of the piece of paper
Let´s assume that
L > WThe Volume (V) of a cylinder is:V(c) = π×r²×h where r is the radius of the circular base, and h is the height of the cylinderFor the first cylinder (V₁) (the one which h = w then r = (L/2)V₁ = π×(L/2)²×w
For the second cylinder (V₂) ( the one with h = L and r = (W/2)V₂ = π×(W/2)²×LThe relation
V₂/V₁ is:V₂/V₁ = π×(W/2)²×L/π×(L/2)²×wV₂/V₁ = W/LBut L > W then V₂/V₁ < 1 or V₂ < V₁The volume of the first cylinder is bigger than the second one:
n:
Hope this helped :]
Tricia deposits $1500 into a savings account that pays 1.2% annual interest compounded quarterly. Write a function to represent the balance A in the account after t years. B. What will be the balance after 3 years C. What will be the balance after 6 years
A. The function to represent the balance in the account after t years is:
[tex]A = 1500(1 + (0.012 / 4))^{(4t)}[/tex]
B. The balance in the account after 3 years will be $1566.66.
C. The balance in the account after 6 years will be $1639.98.
What is Compound interest?Compound interest is defined as interest paid on the original principal and the interest earned on the interest of the principal.
A = P(1+r/100)ⁿ
A. The function to represent the balance in the account after t years can be represented as:
[tex]A = 1500(1 + (0.012 / 4))^{(4t)}[/tex]
where A is the balance in the account after t years, 1500 is the initial deposit, 0.012 is the annual interest rate as a decimal, 4 is the number of compounding periods per year, and (0.012 / 4) is the interest rate per compounding period.
B. To find the balance after 3 years:
A = 1500(1 + (0.012 / 4))⁴ˣ³ = 1500 (1 + (0.012 / 4))¹²
A = 1500 × 1.0444
A = 1566.66
So, the balance in the account after 3 years will be $1566.66.
C. To find the balance after 6 years:
A = 1500 (1 + (0.012 / 4))⁴ˣ⁶ = 1500 * (1 + (0.012 / 4))²⁴
A = 1500 × 1.0932
A = 1639.98
So, the balance in the account after 6 years will be $1639.98.
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Exercise 2.38. We choose one of the words in the following sentence uniformly at random and then choose one of the letters of that word, again uniformly at random: SOME DOGS ARE BROWN (a). Find the probability that the chosen letter is R. (b). Let X denote the length of the chosen word. Determine the probability mass function of X. (c). For each possible value k of X determine the conditional probability P(X k|X 3) Hint. The decomposition idea works just as well for conditional probabilities: if (B\, , Bn} is a partition of 2, then
n P(A| D) = ∑ P(ABk | D). k=1 (d). Determine the conditional probability P(the chosen letter is R | X > 3). (e). Given that the chosen letter is R, what is the probability that the chosen word was BROWN?
a) The probability that the chosen letter is R is 1/10.
b) The probability mass function of X is P(X = 3) = 1/5, P(X = 4) = 2/5, P(X = 5) = 2/5
c) If k = 3 , then P(X = 3 | X = 3) = 1. Otherwise, P(X = k | X = 3) = 0
d) The conditional probability P(the chosen letter is R | X > 3) is 1/11.
e) Given that the chosen letter is R, the probability that the word was BROWN is 1/2.
a) The probability that the chosen letter is R is equal to the number of R's divided by the total number of letters in the sentence. There are two R's in the sentence "SOME DOGS ARE BROWN", so the probability of choosing an R is 2/20 = 1/10.
b) To find the probability mass function of X, we need to find the probability of choosing a word of a certain length. There are five words in the sentence, one of length 3, two of length 4, and two of length 5.
The probability of choosing a word of length 3 is 1/5, the probability of choosing a word of length 4 is 2/5, and the probability of choosing a word of length 5 is 2/5. So, the probability mass function of X is:
P(X = 3) = 1/5
P(X = 4) = 2/5
P(X = 5) = 2/5
c) To find the conditional probability P(X = k | X = 3), we need to find the probability of choosing a word of length k given that we know the word has length 3. If X = 3, then the word must be "DOG". So, the only possibility is P(X = 3 | X = 3) = 1.
For all other values of k, P(X = k | X = 3) = 0 because the word must have length 3 given that X = 3.
d) To find the conditional probability P(the chosen letter is R | X > 3), we need to find the probability of choosing an R given that we know the length of the word is greater than 3. The number of R's in words with length greater than 3 is 1, and the number of letters in these words is 11, so the probability is 1/11.
e) Given that the chosen letter is R, the probability that the word was BROWN is equal to the number of R's in the word BROWN (1) divided by the number of R's in the sentence (2). So, the probability is 1/2.
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What is the range of the relation in the table below?
-2
-1
0
1
2