There are 3 bags each containing 100 marbles. Bag 1 has 75 red and 25 blue marbles. Bag 2 has 60 red and 40 blue marbles. Bag 3 has 45 red and 55 blue marbles. Now a bag is chosen at random and a marble is also picked at random. 1) What is the probability that the marble is blue? 2) What is the probability that the marble is blue when the first bag is chosen with probability 0.5 and other bags with equal probability each? Make sure to clearly define your probabilistic events and mathematically show how different probability laws and rules that you learned in class could be applied to solve the problems.

Answers

Answer 1
Answer:

Answer:

0.4 ; 0.6125

Step-by-step explanation:

Given the following :

Bag 1 : 75 red ; 25 blue

Bag 2: 60 red ; 40 blue

Bag 3: 45 red ; 55 blue

Probability = (required outcome / Total possible outcomes)

A) since the probability of choosing each bag is equal :

BAG A:

P(choosing bag A) = 1 / total number of bags = 1/3 ; P(choosing blue marble) = number of blue marbles / total number of marbles = 25/100

HENCE, choosing a blue marble from bag A : = (1/3 × 75/100) = 25/300

BAG B:

P(choosing bag B) = 1/3 ;

P(choosing blue marble) = number of blue marbles / total number of marbles = 40/100

HENCE, choosing a blue marble from bag A : = (1/3 × 40/100) = 40/300

BAG C:

P(choosing bag C) = 1/3

P(choosing blue marble) = number of blue marbles / total number of marbles = 55/100

HENCE, choosing a blue marble from bag A : = (1/3 × 55/100) = 55/300

= (25/300) × (40/300) × (55/300) = (25 + 40 + 55)/300 = 120/300 = 0.4

2) What is the probability that the marble is blue when the first bag is chosen with probability 0.5 and other bags with equal probability each?

BAG A:

P(choosing bag A) = 0.5 ; P(choosing blue marble) = number of blue marbles / total number of marbles = 25/100

HENCE, choosing a blue marble from bag A : = (0.5 × 75/100) = (0.5 * 0.75) = 0.375

BAG B:

P(choosing bag B) = (1-0.5) / 2 = 0.25 ;

P(choosing blue marble) = number of blue marbles / total number of marbles = 40/100

HENCE, choosing a blue marble from bag A : = (0.25 × 40/100) = (0.25 × 0.4) = 0.1

BAG C:

P(choosing bag C) = (1 - (0.5+0.25)) = 0.25

P(choosing blue marble) = number of blue marbles / total number of marbles = 55/100

HENCE, choosing a blue marble from bag A : = (0.25 × 55/100) = 0.25 × 0.55 = 0.1375

= 0.1375 + 0.1 + 0.375 = 0.6125


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Find the prime factorization of
72

Answers

Answer:

72 = {2,2,2,3,3}

Step-by-step explanation:

72

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Find the radius and height of a cylindrical soda can with a volume of 256cm^3 that minimize the surface area.B: Compare your answer in part A to a real soda can, which has a volume of 256cm^3, a radius of 2.8 cm, and a height of 10.7 cm, to conclude that real soda cans do not seem to have an optimal design. Then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface area of the top and bottom are now twice their values in part A.

B: New radius=?

New height=?

Answers

Answer:

A) Radius: 3.44 cm.

Height: 6.88 cm.

B) Radius: 2.73 cm.

Height: 10.92 cm.

Step-by-step explanation:

We have to solve a optimization problem with constraints. The surface area has to be minimized, restrained to a fixed volumen.

a) We can express the volume of the soda can as:

V=\pi r^2h=256

This is the constraint.

The function we want to minimize is the surface, and it can be expressed as:

S=2\pi rh+2\pi r^2

To solve this, we can express h in function of r:

V=\pi r^2h=256\n\nh=(256)/(\pi r^2)

And replace it in the surface equation

S=2\pi rh+2\pi r^2=2\pi r((256)/(\pi r^2))+2\pi r^2=(512)/(r) +2\pi r^2

To optimize the function, we derive and equal to zero

(dS)/(dr)=512*(-1)*r^(-2)+4\pi r=0\n\n(-512)/(r^2)+4\pi r=0\n\nr^3=(512)/(4\pi) \n\nr=\sqrt[3]{(512)/(4\pi) } =\sqrt[3]{40.74 }=3.44

The radius that minimizes the surface is r=3.44 cm.

The height is then

h=(256)/(\pi r^2)=(256)/(\pi (3.44)^2)=6.88

The height that minimizes the surface is h=6.88 cm.

b) The new equation for the real surface is:

S=2\pi rh+2*(2\pi r^2)=2\pi rh+4\pi r^2

We derive and equal to zero

(dS)/(dr)=512*(-1)*r^(-2)+8\pi r=0\n\n(-512)/(r^2)+8\pi r=0\n\nr^3=(512)/(8\pi) \n\nr=\sqrt[3]{(512)/(8\pi)}=\sqrt[3]{20.37}=2.73

The radius that minimizes the real surface is r=2.73 cm.

The height is then

h=(256)/(\pi r^2)=(256)/(\pi (2.73)^2)=10.92

The height that minimizes the real surface is h=10.92 cm.

Final answer:

The minimal surface area for a cylindrical can of 256cm^3 is achieved with radius 3.03 cm and height 8.9 cm under uniform thickness, and radius 3.383 cm and height 7.14 cm with double thickness at top and bottom. Real cans deviate slightly from these dimensions possibly due to practicality.

Explanation:

For a cylinder with given volume, the surface area A, radius r, and height h are related by the formula A = 2πrh + 2πr^2 (if the thickness is uniform) or A = 3πrh + 2πr^2 (if the top and bottom are double thickness). By taking the derivative of A w.r.t r and setting it to zero, we can find the optimal values that minimize A.

For a volume of 256 cm^3, this gives us r = 3.03 cm and h = 8.9 cm with uniform thickness, and r = 3.383 cm and h = 7.14 cm with double thickness at the top and bottom. Comparing these optimal dimensions to a real soda can (r = 2.8 cm, h = 10.7 cm), we see that the real can has similar but not exactly optimal dimensions. This may be due to practical considerations like stability and ease of holding the can.

Learn more about Optimal Dimensions here:

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What is the factors for 14 and 6

Answers

Answer:

14= 1, 2, 7, 14

6= 1, 2, 3, 6

What is the greatest common factor of 5x^6+35x^4+15x^3

Answers

5 will go into everything. That is one of the factors.

x^3 is in everything as well.

The highest common factor is 5x^3

(5x^6)/(5x^3) = x^(6-3) = x^3

(35x^4)/(5x^3) = 7x^(4-3) = 7x

(15x^3)/(5x^3) = 3*x^(3-3) = 3*1 = 3

So the greatest common factor (determined by the last term) is 5x^3

And the left over polynomial is x^3 + 7x + 3

Square root of 20 is it rational or irrational ?square root of 24 is it rational or irrational ?
square root of 61 is it rational or irrational ?
square root of 62 is it rational or irrational ?
square root of 101 is it rational or irrational ?
square root of 105 is it rational or irrational ?

Answers

Each square root of a number that does not generated as a square of an integer is irrational. This means that none of the given numbers is rational, that is,  they are all irrational.

Good luck!!!!

Suppose that there are 27 sophomores and 14 juniors in a Physiology class. Two students are chosen at random to participate in a class demonstration. What is theprobability of choosing two students at random and the first is a junior and the second one is a sophomore? Round your answer to four decimal places, if necessary.

Answers

The answer would be 7