Let a, b, c, d, e, f be integers selected from the set {1, 2, . . . , 100}, uniformly and at random with
replacement. Set
M = a + 2b + 4c + 8d + 16e + 32f.
What is the expected value of the remainder when M is divided by 64?

Answers

Answer 1

So the expected value of the remainder when M is divided by 64 is approximately 0.2080.

To find the expected value of the remainder when M is divided by 64, we can find the probability that M is congruent to each possible remainder modulo 64 and then sum these probabilities multiplied by the corresponding remainders.

For a positive integer n, let P(n) denote the probability that a positive integer chosen from the set {1, 2, . . . , 100} is congruent to n modulo 64. Then

P(0) = P(64) = P(128) = ... = 1/100

P(1) = P(65) = P(129) = ... = 1/100

P(63) = P(127) = P(191) = ... = 1/100

Since the integers a, b, c, d, e, and f are chosen independently and uniformly at random from the set {1, 2, . . . , 100}, the probability that M is congruent to n modulo 64 is P(n)^6.

Therefore, the expected value of the remainder when M is divided by 64 is

E = 0 * P(0)^6 + 1 * P(1)^6 + 2 * P(2)^6 + ... + 63 * P(63)^6

= (0^6 + 1^6 + 2^6 + ... + 63^6) * P(0)^6

= (0 + 1 + 2 + ... + 63) * P(0)^6

= 2080 * (1/100)^6

= 0.2080

So the expected value of the remainder when M is divided by 64 is approximately 0.2080.

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Answer 2

For a positive integer n, let P(n) denote the probability that a positive integer chosen from the set {1, 2, . . . , 100} is congruent to n modulo 64. Then

P(0) = P(64) = P(128) = ... = 1/100

P(1) = P(65) = P(129) = ... = 1/100

P(63) = P(127) = P(191) = ... = 1/100

Since the integers a, b, c, d, e, and f are chosen independently and uniformly at random from the set {1, 2, . . . , 100}, the probability that M is congruent to n modulo 64 is P(n)^6.

Therefore, the expected value of the remainder when M is divided by 64 is

E = 0 * P(0)^6 + 1 * P(1)^6 + 2 * P(2)^6 + ... + 63 * P(63)^6

= (0^6 + 1^6 + 2^6 + ... + 63^6) * P(0)^6

= (0 + 1 + 2 + ... + 63) * P(0)^6

= 2080 * (1/100)^6

= 0.2080


Related Questions

Ryan has 4 rap songs, 8 country songs, and 2 rock songs.  what is the probability of Ryan picking a rock song or a Pop song

Answers

Answer:

Step-by-step explanation:

Total songs = 8 + 4 + 2 = 14 songs

number of rock songs = 2

Number of rap songs = 4

Probability of rock songs = 2/14 = 1/7

Probability of rap songs = 4/14 = 2/7

What is the resistance of a coil which draws a current of 200na​

Answers

Answer:

600000Ω

Step-by-step explanation:

1μA = 10⁻⁶ A

200μA = 0.0002 A

Potential difference = V = 120 V

Current = I = 0.0002 A

Resistance = R

R = V / I

  = 120 / 0.0002

  = 600000Ω

I NEED HELP ASAPPPPP ​

Answers

Answer:

y=55 degrees x=125 degrees

Step-by-step explanation:

Find Y.
Round to the nearest tenth.

Answers

Answer:

Y ≈ 59.0°

Step-by-step explanation:

Using the Cosine rule

cosY = [tex]\frac{x^2+z^2-y^2}{2xz}[/tex]

with x = 22, y = 20, z = 18

cosY = [tex]\frac{22^2+18^2-20^2}{2(22)(18)}[/tex]

        = [tex]\frac{484+324-400}{792}[/tex]

        = [tex]\frac{408}{792}[/tex] , then

Y = [tex]cos^{-1}[/tex] ( [tex]\frac{408}{792}[/tex] ) ≈ 59.0° ( to the nearest tenth )

in a class of 38 students, 16 passed chemistry, ​

Answers

Answer:

what are you asking exaclty ???

Step-by-step explanation:

Question 2
Find the Total Surface Area and volume of the prism below. Round to the nearest hundredth, if necessary.
8 cm
16 cm
18 cm
cm squared
Total Surface Area =
cm cubed
Volume =
Back
Next
M

Answers

the volume is 2304
the surface area is 1120

plsss help and please dont be scammer

Answers

Answer:

38 students

Step-by-step explanation:

because 19% of 205 is 38.95, but not fully 39 students, so about 38 students

Any help with this question please?

Consider this equation.
c = ax – bx
Joseph claims that if a, b, and c are non-negative integers, then the equation has exactly one solution for x. Select all cases that show Joseph’s claim is incorrect.
A. a – b = 1, c ≠ 1
B. a = b, c = 0
C. a ≠ b, c = 0
D. a – b = 1, c = 0
E. a = b, c ≠ 0

Answers

Answer:

B.a=b, c≠0

C.a=b, c=0

D.a-b=1, c≠1

Step-by-step explanation:

The equation given is c = ax - bx. We can factor the right-hand side to obtain an equivalent equation which is c = (a-b)x

Let’s explore each answer choice given. We are looking for cases where there is no one solution for the equation.

A

a-b = 1 so the right-hand side becomes 1x and we have x=c. Since c is 0 we have one solution that is x=0

B

a=b so a-b =0 and the equation becomes 0=c but the answer choice says c does not equal zero. So in this case there is no solution. This is a correct answer to the problem.

C

This is the same as choice B but since C =0 both sides of the equation equal zero. We get 0=0 but notice that this is true no matter what the value of x is so this equation is called identity and any value of x will do so there isn’t one solution but rather infinitely many. This is another right answer.

D

Here a-b=1 so we end up with x = c and since c doesn’t equal one any value of x except 1 is a solution so there isn’t one solution but infinitely many. This too is an answer to the question.

E

Since a doesn’t equal b and since c = 0 we have (a-b)x = 0 so. Either a-b is zero but since a and b are different this can’t be or x is zero. This there is one solution: x=0.

From the above, the answer to the question is choices B, C, and D

TANIA WANTS TO BUY STICKERS OVER THE INTERNET. EACH STICKER COSTS $2 AND SHIPPING THE ENTIRE ORDER COSTS $6. IF SHE CAN SPEND NO MORE THAN $30, HOW MANY STICKERS CAN SHE BUY? WRITE AND SOLVE AN INEQUALITY SHOWING THIS SITUATION. CHECK THE IN THE Soultion in the inequality. ​

Answers

She can buy 12 stickers

Erik saves $279 in one week, $558 in two weeks, and $837 in three weeks. How much will Erik
have saved after 6 weeks?

Answers

1,674 dollars after saved 6 weeks, your welcome

• A rope 75 meters long was cut into pieces that were each 1.5 meters long.
How many pieces resulted?
Help!!

Answers

The answer is 50.
75/1.5=50

Can I please have brainliest?

guys help me lol plssss

Answers

Answer:

ITS RED

Step-by-step explanation:

Answer:

the red one/ fgh=vwh

Step-by-step explanation:

No
© A
A cube
side 4cm has the same
as a cone with diameter 7cm. calculate
correct to the nearest cm,
the
گج
the height
heig
cone,​

Answers

The answer is A. to your question

I dont understand.i need help​

Answers

Answer:

A= 160 in. squared

Step-by-step explanation:

Vector u has initial point at (3, 9) and terminal point at (–7, 5). Vector v has initial point at (1, –4) and terminal point at (6, –1).
What is u + v in component form?
⟨-10, -4⟩
⟨-5, -1⟩
⟨3, 9⟩
⟨5, 3⟩

Answers

Answer:

⟨-5, -1⟩

Step-by-step explanation:

Vector:

A vector is given by its endpoint subtracted by its initial point.

Vector u has initial point at (3, 9) and terminal point at (–7, 5)

Then

[tex]u = (-7, 5) - (3,9) = (-7 - 3, 5 - 9) = (-10,-4)[/tex]

Vector v has initial point at (1, –4) and terminal point at (6, –1).

Then

[tex]v = (6,-1) - (1,-4) = (6-1,-1-(-4)) = (5,3)[/tex]

What is u + v in component form?

[tex]u + v = (-10,-4) + (5,3) = (-10+5,-4+3) = (-5,-1)[/tex]

⟨-5, -1⟩ is the answer.

A share of stock LOSES value in each week of a month. In which quadrant would the data points for this situation be graphed? ( The 2 variables are WEEKS and STOCK PRICE)

Answers

Answer:

Quad III

Step-by-step explanation:

III.

When calculating simple probability,
what goes in the numerator and what
goes in the denominator?
Please.

Answers

Answer:

The top number in a fraction. Shows how many parts we have. (The bottom number is the Denominator and shows how many equal parts the item is divided into.)

Step-by-step explanation:

What's the measure of X

Answers

the triangles are both ratios of each other so the angles are the same

180-25-85= 70°

the answer is A 70°

Enter the value of (0.45) divided by 9/10

Answers

The value is .5 or 1/2

A baker has 6 pounds of sugar. He divides into 3 containers. He then uses 1 container to bake pies. Which expression shows how many pounds of sugar the baker used?

Answers

Step-by-step explanation:

You have 6 pounds of sugar, when you set it into 3 containers, you will get:

6:3=2 ( pounds per container)

And he used one container for pies it would be:

1 container contains 2 pounds of sugar, so he used 2 pounds of sugar.

In a 30-60-90 triangle, if the the side opposite 30 degrees has a length
of 5, what is the length of the hypotenuse?

Answers

Might Want to draw this and Its very long but Hope it helped :)

Answer:

The sides are all of the same length - let's say a. The angles are all the same too, and since the angles must add up to 180∘, we conclude that the three angles in the equilateral triangle are equal to 180∘/3=60∘.

Now we do something sneaky. We draw a line all the way down from the top vertex of the triangle to the midpoint of the bottom line.

This new line cuts our equilateral triangle in half. What are the angles in one half?

The angle at the bottom is 90∘.

One of the angles is the same as one of the angles in the original equilateral triangle, so it is 60∘.

So the third angle must be 180∘−90∘−60∘=30∘.

Now the hypotenuse of this new triangle is a, the side length of the equilateral triangle. And the length of the shortest side is a/2, since the line we drew cut the bottom line in half.

Marya has $146 in her checking account . She writes a check for $193 . What is the new balance in her checking account ?

Answers

Answer:

-$47

Step-by-step explanation:

If she has $146 in he account and we take away $193, we are left with -$47, making her in debt.

Johnathan wants to paint the bottom of his swimming pool the color blue. the swimming pool has a diameter of 50 feet . How many square feet will Johnathan need to paint ?

Answers

Johnathan will need to paint 1,962.5 square feet.

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Find the area of the shape below

Answers

Answer:

I have questions in physics if you know it answer it

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Answers

Answer:no

Step-by-step explanation:

the (x) fators out more than once

a sales clerk wants to make at least $519 . If he makes a base salary of $335 and $10.25 per hour at his part time job how many hours will he have to work?

Answers

Answer:

18 hours at least to reach the goal

Step-by-step explanation:

Well 519 is the lowest objective so lets aim for that

we can make an equation

since base pay is 335 that doesn’t change

We can make x for hours

335+10.25x ≥ 519

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Help please!!!! I’ll appreciate it so much!

Answers

Question 1) x + 6 = 20
x + 6 - 6 = 20 - 6
x = 14

Question 2) x - 2 = 5
x - 2 + 2 = 5 + 2
x = 7

Question 3) 15 + m = 9
15 - 15 + m = 9 - 15
m = -6

Find the scale factor used to go from the actual picture to the scale drawing!

Answers

Answer:

1/3

Step-by-step explanation:

9:6

3:2

9/3 = 3

6/2 = 3

9 x (1/3) =  3

6 x (1/3) = 2

scale factor is 1/3

Best Company sells personalized mugs for $18 each plus a fixed fee of $26 for shipping and handling for the entire order. The Cup Expert Company sells personalized mugs for $22 each plus a fixed fee for $14 for shipping and tandling for the entire orderHow many personalized mugs would a customer have to purchase for the total cost of each company to be the same?

Answers

Answer:

3

Step-by-step explanation:

Write each sentence as an equation.
1 .Two less than a number n is fifty two.
2. Three less than a number is twelve.
3. The sum of five and a number is ten less than fifty.
4. Three more than a number is sixty.
5. The sum of x and 6 is two less than ten,
6. A number minus 9 is twelve.
7. The sum of a number and five is thirteen.
8. The sum of x and ten is twenty.
9. A number five times is twenty five.
10. Twice a number is two more than five,

Answers

Answer:

n - 2 = 52x - 3 = 125 + x = 50 - 103 + x = 60x + 6 = 10 - 2x - 9 = 12x + 5 = 13x + 10 = 205x = 252x = 5 + 2

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have a great day!!

The answer is given,

1.n - 2 = 52

2.x - 3 = 12

3.5 + x = 50 - 10

4.3 + x = 60

5.x + 6 = 10 - 2

6.x - 9 = 12

7.x + 5 = 13

8.x + 10 = 20

9.5x = 25

10.2x = 5 + 2

What is equation?

An equation is a  mathematical statement that is made up of two expressions connected by an equal sign.

here, we have,

as the description given,

we get,

1.n - 2 = 52

2.x - 3 = 12

3.5 + x = 50 - 10

4.3 + x = 60

5.x + 6 = 10 - 2

6.x - 9 = 12

7.x + 5 = 13

8.x + 10 = 20

9.5x = 25

10.2x = 5 + 2

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