(A) Over 1000 students organized to celebrate running water and electricity. To count the exact number of students protesting, the chief organizer lined the students up in columns of different length. If the students are arranged in columns of 3, 5, and 7, then 2, 3, and 4 people are left out, respectively. What is the minimum number of students present? Solve it with Chinese Remainder Theorem. (B) Prove that for n> 1, if 935 = 5 x 11 x 17 divides n80 – 1, then 5, 11, and 17 do not divide n.

Answers

Answer 1

Solution :

A). x = 2 (mod 3)        [tex]$\mu = 3\times 5 \times 7 = 105$[/tex]

x = 3 (mod 5)       [tex]$y_1=35^{-1} (\mod 3)$[/tex]

x = 4 (mod 7)        [tex]y_1=2[/tex]

[tex]$y_2=21^{-1}(\mod5) = 1$[/tex]

[tex]$y_3=15^{-1}(\mod7) = 1$[/tex]

[tex]$x=2 \times 35 \times 2 + 3\times 21\times 1+4\times 15\times 1$[/tex]

   [tex]=140+63+60[/tex]

   [tex]=263[/tex]

   ≡ 53(mod 105)

Hence the solution is 105 k + 53 > 1000 for k = 10

Therefore, the minimum number of students = 1103

B). [tex]$\phi (935) = 640$[/tex]

By  Eulu's theory

[tex]$935 | a^{640}_n -1$[/tex] if n and 935 are coprime.

Now, [tex]$935|n^{80}-1$[/tex]  and 80 x 8 = 640

[tex]$935|n^{640}-1$[/tex]   ⇒  g(n,935) = 1

                     ⇒ 5, 11, 17 do not divide n


Related Questions

Prove that A.M, G.M. and H.M between any two unequal positive numbers satisfy the following relations.
i. (G.M)²= (A.M)×(H.M)
ii.A.M>G.M>H.M​

Answers

Answer:

See below

Step-by-step explanation:

we want to prove that A.M, G.M. and H.M between any two unequal positive numbers satisfy the following relations.

(G.M)²= (A.M)×(H.M) A.M>G.M>H.M

well, to do so let the two unequal positive numbers be [tex]\text{$x_1$ and $x_2$}[/tex] where:

[tex] x_{1} > x_{2}[/tex]

the AM,GM and HM of [tex]x_1[/tex] and[tex] x_2[/tex] is given by the following table:

[tex]\begin{array}{ |c |c|c | } \hline AM& GM& HM\\ \hline \dfrac{x_{1} + x_{2}}{2} & \sqrt{x_{1} x_{2}} & \dfrac{2}{ \frac{1}{x_{1} } + \frac{1}{x_{2}} } \\ \hline\end{array}[/tex]

Proof of I:

[tex] \displaystyle \rm AM \times HM = \frac{x_{1} + x_{2}}{2} \times \frac{2}{ \frac{1}{x_{1} } + \frac{1}{x_{2}} } [/tex]

simplify addition:

[tex] \displaystyle \frac{x_{1} + x_{2}}{2} \times \frac{2}{ \dfrac{x_{1} + x_{2}}{x_{1} x_{2}} } [/tex]

reduce fraction:

[tex] \displaystyle x_{1} + x_{2} \times \frac{1}{ \dfrac{x_{1} + x_{2}}{x_{1} x_{2}} } [/tex]

simplify complex fraction:

[tex] \displaystyle x_{1} + x_{2} \times \frac{x_{1} x_{2}}{x_{1} + x_{2}} [/tex]

reduce fraction:

[tex] \displaystyle x_{1} x_{2}[/tex]

rewrite:

[tex] \displaystyle (\sqrt{x_{1} x_{2}} {)}^{2} [/tex]

[tex] \displaystyle AM \times HM = (GM{)}^{2} [/tex]

hence, PROVEN

Proof of II:

[tex] \displaystyle x_{1} > x_{2}[/tex]

square root both sides:

[tex] \displaystyle \sqrt{x_{1} }> \sqrt{ x_{2}}[/tex]

isolate right hand side expression to left hand side and change its sign:

[tex]\displaystyle\sqrt{x_{1} } - \sqrt{ x_{2}} > 0[/tex]

square both sides:

[tex]\displaystyle(\sqrt{x_{1} } - \sqrt{ x_{2}} {)}^{2} > 0[/tex]

expand using (a-b)²=a²-2ab+b²:

[tex]\displaystyle x_{1} -2\sqrt{x_{1} }\sqrt{ x_{2}} + x_{2} > 0[/tex]

move -2√x_1√x_2 to right hand side and change its sign:

[tex]\displaystyle x_{1} + x_{2} > 2 \sqrt{x_{1} } \sqrt{ x_{2}}[/tex]

divide both sides by 2:

[tex]\displaystyle \frac{x_{1} + x_{2}}{2} > \sqrt{x_{1} x_{2}}[/tex]

[tex]\displaystyle \boxed{ AM>GM}[/tex]

again,

[tex]\displaystyle \bigg( \frac{1}{\sqrt{x_{1} }} - \frac{1}{\sqrt{ x_{2}}} { \bigg)}^{2} > 0[/tex]

expand:

[tex]\displaystyle \frac{1}{x_{1}} - \frac{2}{\sqrt{x_{1} x_{2}} } + \frac{1}{x_{2} }> 0[/tex]

move the middle expression to right hand side and change its sign:

[tex]\displaystyle \frac{1}{x_{1}} + \frac{1}{x_{2} }> \frac{2}{\sqrt{x_{1} x_{2}} }[/tex]

[tex]\displaystyle \frac{\frac{1}{x_{1}} + \frac{1}{x_{2} }}{2}> \frac{1}{\sqrt{x_{1} x_{2}} }[/tex]

[tex]\displaystyle \rm \frac{1}{ HM} > \frac{1}{GM} [/tex]

cross multiplication:

[tex]\displaystyle \rm \boxed{ GM >HM}[/tex]

hence,

[tex]\displaystyle \rm A.M>G.M>H.M[/tex]

PROVEN

find the amount of time to the nearest day it would take a deposit of $2500 to grow to $1 million at 2% compounded continuously. find how many days & years

Answers

Answer:

Years = natural log (Total / Principal) / Rate

Years = natural log (1,000,000 / 2,500) / .02

Years = natural log (400) / .02

Years =  5.9914645471 / .02

It would take  299.573227355 Years

Source: http://www.1728.org/rate2.htm

Step-by-step explanation:

In the statements below, V is a vector space. Mark each statement true or false. Justily each answer a. The set R is a two-dimensional subspace of R3.Choose the correct answer below O A. False, because R2 is not closed under vector addition. O B. True, because R2 is a plane in R3 Ос. False, because the set R2 is not even a subset of R3 OD. True, because every vector in R2 can be represented by a linear combination of vectors inR3 b. The number of variables in the equation Ax 0 equa's the dimension of Nul A. Choose the correct answer below O A. False, because the number of free variables is equal to the dimension of Nul A. O B. True, because the number of variables in the equation Ax 0 equals O C. True, because the dimension of Nul A equals the largest any solution to O D. False, because the number of plvot columns is equal to the dimension of Nud A. c. A vector space the number of columns in A and the number of columns in A equa's the dimension of Nul A. number of Os in any solution to the equation Ax -b, and the equation Ax- 0 always has the trivial solution, so the number of variables is infinite-dimensional if it is spanned by an infinite set Choose the correct answer below O A. True, because the dimension of a vector space is equal to the number of elements in a set that spans O B. Faise, because a basis for the vector space may O C. True, because the dimension of a vector space number of O D. Faise, because all vector spaces are finite-dimensional. d. If dim Van and it S spans V, then S is a basis of V. Choose the correct answer below. the vector space. have only finitely many elements, which would make the vector space finite-dimensional is the number of vectors in a basis for that vector space, and a vector space spanned by an infinite set has a basis with an infinite number of vect O A. False, because the set S must have less than n elements O B. True, because if a vector space is finite-dimensional, then a set that spans t is a basis of the vector space O C. False, in order for S to be a basis, it must also have n elements O D. True, because if a set spans a vector space, regardiess of the dimension of the vector space, then that setis a basis of the vector spaoe e. The only three-dimensional subspace of R3 is R3 itself. Choose the correct answer below Faise, because False, because any subspaces of R3 which contain three-element vectors are three-dimensional, but most of these most three-dimensional subspaces of R3 are spanned by a linearly dependent set of tree vectors, but R can only be sparned by thre Inearly independent vectors subspaces do not contain all of R
D. True, because any three linearly dependent vectors in R3 span all of R3, so there is no three-dmensional subspace of R' that is not R

Answers

Answer:

A. False

B. True

C. False

D. True

Step-by-step explanation:

Only three dimensional subspace for R3 is R3 itself. In a 3 d subspace there are 3 basis vectors which are all linearly independent vectors. Dimension of a vector is number of subspace in that vector. Finite set can generate infinite dimension vector space.

Turn 43 1/23 into an improper fraction

Answers

Answer:

990/23

Step-by-step explanation:

Step 1

Multiply the denominator by the whole number

23 × 43 = 989

Step 2

Add the answer from Step 1 to the numerator

989 + 1 = 990

Step 3

Write answer from Step 2 over the denominator

990/23

I hope this answer helps you out! Brainliest would be appreciated.

If y = ax^2 + bx + c passes through the points (-3,10), (0,1) and (2,15), what is the value of a + b + c?

Answers

Hi there!

[tex]\large\boxed{a + b + c = 6}[/tex]

We can begin by using the point (0, 1).

At the graph's y-intercept, where x = 0, y = 1, so:

1 = a(0)² + b(0) + c

c = 1

We can now utilize the first point given (-3, 10):

10 = a(-3)² + b(-3) + 1

Simplify:

9 = 9a - 3b

Divide all terms by 3:

3 = 3a - b

Rearrange to solve for a variable:

b = 3a - 3

Now, use the other point:

15 = a(2)² + 2(3a - 3) + 1

14 = 4a + 6a - 6

Solve:

20 = 10a

2 = a

Plug this in to solve for b:

b = 3a - 3

b = 3(2) - 3 = 3

Add all solved variables together:

2 + 3 + 1 = 6

0.14 converted as a fraction simplest form.​

Answers

Answer: 7 / 50

Step-by-step explanation:

Given

0.14

Convert to 100-denominator fraction

= 14 ÷ 100

= 14/100

Divide both numerator and denominator by 2

=(14 ÷ 2) / (100 ÷ 2)

=7 / 50

Hope this helps!! :)

Please let me know if you have any questions

Help me because I dont understand

Answers

Answer:

105 sq ft + 31 sq ft

Step-by-step explanation:

=  136 sq ft

Hope it helps✌✌

A sample of 45 bottles of soft drink showed a variance of 1.1 in their contents. The process engineer wants to determine whether or not the standard deviation of the population is significantly different from 0.9 ounces. What is the value of the test statistic

Answers

Answer:

The value of the test statistic is 59.75.

Step-by-step explanation:

The test statistic for the population standard deviation is:

[tex]\chi^2 = \frac{n-1}{\sigma_0^2}s^2[/tex]

In which n is the sample size, [tex]\sigma_0[/tex] is the value tested and s is the sample standard deviation.

A sample of 45 bottles of soft drink showed a variance of 1.1 in their contents.

This means that [tex]n = 45, s^2 = 1.1[/tex]

The process engineer wants to determine whether or not the standard deviation of the population is significantly different from 0.9 ounces.

0.9 is the value tested, so [tex]\sigma_0 = 0.9, \sigma_0^2 = 0.81[/tex]

What is the value of the test statistic

[tex]\chi^2 = \frac{n-1}{\sigma_0^2}s^2[/tex]

[tex]\chi^2 = \frac{44}{0.81}1.1 = 59.75[/tex]

The value of the test statistic is 59.75.

Please help me answer this question?

Answers

Answer:

2+2

Step-by-step explanation:

2 + 4!

3-5

3_4

3-6

2-5

2+5

2_3

2-5

Answer:

(A) 12x³ - 12x

(B) -288

(C) y = -288x - 673

(D) x = 0, 1, -1

Step-by-step explanation:

See images. If it's not clear let me know.

Which letter on the diagram below represent a diameter of the circle

Answers

Answer:

where is your diagram?

Step-by-step explanation:

 Solve each system by graphing.

Answers

9514 1404 393

Answer:

  (x, y) = (4, -4)

Step-by-step explanation:

A graphing calculator makes graphing very easy. The attachment shows the solution to be (x, y) = (4, -4).

__

The equations are in slope-intercept form, so it is convenient to start from the y-intercept and use the slope (rise/run) to find additional points on the line.

The first line can be drawn by staring at (0, -2) and moving down 1 grid unit for each 2 to the right.

The second line can be drawn by starting at (0, 2) and moving down 3 grid units for each 2 to the right.

The point of intersection of the lines, (4, -4), is the solution to the system of equations.

f(x)=3(x+5)+4/xwhat is f (a+2) solve this problem with showing the work

Answers

smac my az like the drum

A sample of 375 college students were asked whether they prefer chocolate or vanilla ice cream. 210 of those surveyed said that they prefer vanilla ice cream. Calculate the sample proportion of students who prefer vanilla ice cream.

Answers

Answer:

The sample proportion of students who prefer vanilla ice cream is 0.56.

Step-by-step explanation:

Sample proportion of students who prefer vanilla ice cream:

Sample of 375 students.

Of those, 210 said they prefer vanilla ice cream.

The proportion is:

[tex]p = \frac{210}{375} = 0.56[/tex]

The sample proportion of students who prefer vanilla ice cream is 0.56.

what is b x b equialent to?

Answers

Answer:

b^2

Step-by-step explanation:

You're going to add the exponents from b x b, both carry a 1 in their powers (or exponents)

so b^1 + b^1 = b^2

Answer:

b^2

Step-by-step explanation:

b*b = b^2

2 cans of beans cost 98¢ how many cans can you buy for $3.92?

Answers

You can buy 8 cans for $3.92

0.98 divided by 2 = 0.49
3.92 divided by 0.49 = 8

8x0.49=3.92

Use the given information to find the number of degrees of​ freedom, the critical values χ2L and χ2R​, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution. Nicotine in menthol cigarettes 99​% ​confidence; n=23​, s=0.28 mg.
df = (Type a whole​ number.)
χ2L = ​(Round to three decimal places as​ needed.)
χ2R = ​(Round to three decimal places as​ needed.)
The confidence interval estimate of σ is __ mg < σ < __ mg. ​(Round to two decimal places as​ needed.)

Answers

Answer:

χ²R = 8.643

χ²L = 42.796

0.20 < σ < 0.45

Step-by-step explanation:

Given :

Sample size, n = 23

The degree of freedom, df = n - 1 = 23 - 1 = 22

At α - level = 99%

For χ²R ; 1 - (1 - 0.99)/2= 0.995 ; df = 22 ; χ²R = 8.643

For χ²L ; (1 - 0.99)/2 = 0.005 ; df = 22 ; χ²L = 42.796

The confidence interval of σ ;

s * √[(n-1)/χ²L] < σ < s * √[(n-1)/χ²R)]

0.28 * √(22/42.796) < σ < 0.28 * √(22/8.643)

0.2008 < σ < 0.4467

0.20 < σ < 0.45

Using Eulers formula, how many edges does a polyhedron with 9 faces and 14 vertices have?

Answers

By Euler's Formula

F + V = E + 2

Solution

F = 9

V = 14

E = ?

Substuting the values

⇨ 9 + 14 = E + 2

⇨ 23 = E + 2

⇨ 23 - 2 = E

⇨ 21 = E

Hence , the number of edges in polyhedron is 21.

The number of edges of a polyhedron with 9 faces and 14 vertices have will be 21.

What is a polygon?

The polygon is a 2D geometry that has a finite number of sides. And all the sides of the polygon are straight lines connected to each other side by side.

here, we have,

Using Euler's formula, the number of the edges does a polyhedron with 9 faces and 14 vertices have

We know the formula for the edges of the polyhedron will be

By Euler's Formula

F + V = E + 2

The number of faces, vertices, and edges of a polyhedron are denoted by the letters F, V, and E.

Then we have

Solution

F = 9

V = 14

E = ?

Substuting the values

⇨ 9 + 14 = E + 2

⇨ 23 = E + 2

⇨ 23 - 2 = E

⇨ 21 = E

Hence , the number of edges in polyhedron is 21.

More about the polygon link is given below.

brainly.com/question/17756657

#SPJ2

Simplify the given expression.

Answers

Answer:

8x-21

----------------------

(2x-7)(2x+7)

Step-by-step explanation:

7                       4

-----------   + ------------

4x^2 -49    2x+7

Factor  ( notice that it is the difference of squares)

7                       4

-----------   + ------------

(2x)^2 - 7^2    2x+7

7                       4

-----------       + ------------

(2x-7)(2x+7)    2x+7

Get a common denominator

7                       4(2x-7)

-----------       + ------------

(2x-7)(2x+7)    (2x-7)(2x+7)

Combine

7 +4(2x-7)

----------------------

(2x-7)(2x+7)  

7 +8x-28

----------------------

(2x-7)(2x+7)  

8x-21

----------------------

(2x-7)(2x+7)  

Answer:

(8x - 21) / (2x + 7)(2x - 7)

Step-by-step explanation:

7 / (4x^2 - 49)+ 4 / (2x + 7)

= 7 / (2x + 7)(2x - 7) + 4 / (2x + 7)

LCM = (2x + 7)(2x - 7)   so we have

(7 + 4(2x - 7) / (2x + 7)(2x - 7)

=   (8x - 21) / (2x + 7)(2x - 7).

The distance from the green point on the parabola to the parabolas focus is 11. What is the distance from green point to the directrix?

Answers

Answer:

answer 11

Step-by-step explanation:

I think it the right answer

PLEASE HELPPPPPPPPPPPPPP

Answers

Answer:

False

Step-by-step explanation:

To find the inverse of a function, switch the variables and solve for y.

The inverse of f(n)=-(n+1)^3:

[tex]y=-(n+1)^3[/tex]

[tex]n=-(y+1)^3[/tex]

[tex]\sqrt[3]{n} =-(y+1)[/tex]

[tex]\sqrt[3]{n} =-y-1[/tex]

[tex]\sqrt[3]{n} +1=-y[/tex][tex]-(\sqrt[3]{n} +1)=y[/tex]

[tex]-\sqrt[3]{n} -1=y[/tex]

Answer:

False

Step-by-step explanation:

Diane must choose a number between 49 and 95 that is a multiple of 2, 3, and 9. Write all the numbers that she could choose. If
there is more than one number, separate them with commas?

Answers

The set of numbers that Diane can choose is:

{54, 60, 66, 72, 78, 84, 90}

Finding common multiples of 2, 3, and 6:

A number is a multiple of 2 if the number is even.

A number is a multiple of 3 if the sum of its digits is multiples of 3.

A number is a multiple of 6 if it is a multiple of 2 and 3.

Then we only need to look at the first two criteria.

First, let's see all the even numbers in the range (49, 95)

These are:

{50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94}

All of these are multiples of 2.

Now we need to see which ones are multiples of 3.

To do it, we sum its digits and see if that sum is also a multiple of 3.

50: 5 + 0 = 5 this is not multiple of 3.

52: 5 + 2 = 7 this is not multiple of 3.

54: 5 + 4 = 9 this is multiple of 3, so 54 is a possible number.

And so on, we will find that the ones that are multiples of 3 are:

54: 5 + 4 = 9.

60: 6 + 0 = 6

66: 6 + 6 = 12

72: 7 + 2 = 9

78: 7 + 8 = 15

84: 8 + 4 = 12

90:9 + 0 = 9

Then the numbers that Diane could choose are:

{54, 60, 66, 72, 78, 84, 90}

If you want to learn more about multiples, you can read:

https://brainly.com/question/1553674

in a survey of 90 students, the ratio of those who work outside the home to those who don't is 6:4. How many students work outside the home according to this survey? SHOW ALL WORK! AND ONLY ANSWER IF YOU KNOW THE ANSWER!

Answers

9514 1404 393

Answer:

  54

Step-by-step explanation:

The fraction of the total that work outside the home is ...

  outside/(outside +inside) = 6/(6+4) = 6/10

Then the number of those surveyed who work outside the home is ...

  (6/10)(90) = 54 . . . work outside the home

Let V be the volume of the solid obtained by rotating about the y-axis the region bounded y = sqrt(25x) and y = x^2/25. Find V by slicing & find V by cylindrical shells.

Answers

Explanation:

Let [tex]f(x) = \sqrt{25x}[/tex] and [tex]g(x) = \frac{x^2}{25}[/tex]. The differential volume dV of the cylindrical shells is given by

[tex]dV = 2\pi x[f(x) - g(x)]dx[/tex]

Integrating this expression, we get

[tex]\displaystyle V = 2\pi\int{x[f(x) - g(x)]}dx[/tex]

To determine the limits of integration, we equate the two functions to find their solutions and thus the limits:

[tex]\sqrt{25x} = \dfrac{x^2}{25}[/tex]

We can clearly see that x = 0 is one of the solutions. For the other solution/limit, let's solve for x by first taking the square of the equation above:

[tex]25x = \dfrac{x^4}{(25)^2} \Rightarrow \dfrac{x^3}{(25)^3} = 1[/tex]

or

[tex]x^3 =(25)^3 \Rightarrow x = \pm25[/tex]

Since we are rotating the functions around the y-axis, we are going to use the x = 25 solution as one of the limits. So the expression for the volume of revolution around the y-axis is

[tex]\displaystyle V = 2\pi\int_0^{25}{x\left(\sqrt{25x} - \frac{x^2}{25}\right)}dx[/tex]

[tex]\displaystyle\:\:\:\:=10\pi\int_0^{25}{x^{3/2}}dx - \frac{2\pi}{25}\int_0^{25}{x^3}dx[/tex]

[tex]\:\:\:\:=\left(4\pi x^{5/2} - \dfrac{\pi}{50}x^4\right)_0^{25}[/tex]

[tex]\:\:\:\:=4\pi(3125) - \pi(7812.5) = 14726.2[/tex]

The maximum and minimum Values of a quadratic function are called as______of the function.

Answers

Answer:

the answer is B ...Extreme Values

In 1815, Sophie Germain won a mathematical prize given by the Institut de France for her work on the theory of elasticity. The prize was a medal made of 1 kilogram of gold. How much is the medal worth today in U.S. dollars and in euros

Answers

Answer:

gold price : $58.72/gram

$58,720 per kilo(1000) grams

Step-by-step explanation:

look at the image below

Answers

Answer:

SA = 153.9m^2

Step-by-step explanation:

SA = 4[tex]\pi[/tex][tex]r^{2}[/tex]

r = 3.5

SA = 4[tex]\pi[/tex][tex](3.5)^{2}[/tex]

SA = 4[tex]\pi[/tex](12.25)

SA = 49[tex]\pi[/tex]

SA = 153.9m^2

(x)=4log(x+2) Which interval has the smallest average rate of change in the given function? 1≤x≤3 5≤x≤7 3≤x≤5 −1≤x≤1

Answers

Answer:

5≤x≤7

Step-by-step explanation:

For a given function f(x), the average rate of change in a given interval:

a ≤ x ≤ b

is given by:

[tex]r = \frac{f(b) - f(a)}{b - a}[/tex]

Here we have:

f(x) = 4*log(x + 2)

And we want to see which interval has the smallest average rate of change, so we just need fo find the average rate of change for these 4 intervals.

1)  1≤x≤3

here we have:

[tex]r = \frac{f(3) - f(1)}{3 - 1} = \frac{4*log(3 + 2) - 4*log(1 + 2)}{2} = 0.44[/tex]

2)  5≤x≤7

[tex]r = \frac{f(7) - f(5)}{7 - 5} = \frac{4*log(7 + 2) - 4*log(5 + 2)}{2} = 0.22[/tex]

3) 3≤x≤5

[tex]r = \frac{f(5) - f(3)}{5 - 3} = \frac{4*log(5 + 2) - 4*log(3 + 2)}{2} = 0.29[/tex]

4) −1≤x≤1

[tex]r = \frac{f(1) - f(-1)}{1 - (-1)} = \frac{4*log(1 + 2) - 4*log(-1 + 2)}{2} = 0.95[/tex]

So we can see that the smalles average rate of change is in 5≤x≤7

write your answer as an integer or as a decimal rounded to the nearest tenth​

Answers

Answer:Mark Brainliest please

Answer is 4.86 which is rounded to 5

Step-by-step explanation:

Cos 40 degree = VW/7

0.694 =VW/7

0.694 * 7 =VW

4.858 =VW

VW=4.86 is the answer

A local food bank uses volunteers to staff the kitchen. If there are 30 college students working there out of a total of 100 volunteers, what is the probability that in a sample of 10 volunteers, 4 of them are college students? Four decimal places please!

Answers

When you form a 10 people sample from a pool of large amount with equal probabilities, let s denote student and n denote non student, each spot in the sample has 1/2 chance to be s or n so each event has 1/2*1/2*1/2…. =(1/2)^10 probability. For example: snsnsnnnns. And intuitively you should see that there have a permutation 2^10 = 1024 ways to line them up (permutation with repetition like a suitcase 3-line passcode lock with 10 numbers on each line, you will have exactly 10*10*10 ways to choose passcodes, which is 10^3=1000 ways) , and each way has a probability a equal probability which is 1/1024 which is (1/2)^10.

Now let’s pick 4 students that we know for sure, mark 10 spots and have them walk randomly into each taking exactly 1 spot. There are C (10,4) possible combinations of choosing 4 people to walk in, for example 4s taking spot 3,6,7,10; 2,5,7,8; … C(10,4) = 210 ways. This represents that in a 10 spot line with 4 students and 6 non students in it, no matter how they are arranged there are 210 ways they line up. Now since each way has a prob 1/1024 we have a probability of 210 ways multiply by 1/1024 which = 55/256.

That was the case with large pool of equal number of S and N let’s make it a more realistic say 50 students and 50 non students. (1/2 derived from 50/100 or 5/10)

So now we have 30 students and 70 non students pool, the answer should be (3/10)^4*(7/10) ^6 which is the probability of 4 students and 6 non students lined up in some way while choosing them from the pool. Here we can also see that each 10 people sample no longer have the same probability. And that number came out to be 0.0081*0.117649=0.00095. And again there are 210 ways of arranging them so the probability is 210 ways multiply by 0.00095 which is 20%. This is the Answer.

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