Mathematics High School

In a bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of all total. Of their output 5,4, 2 percent are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machines A, B and C.

Answers

Answer 1

The probability that the defective bolt was manufactured by machine A is approximately 0.4386, by machine B is approximately 0.4930, and by machine C is approximately 0.0702.

To find the probability that the defective bolt was manufactured by each machine, we can use Bayes' theorem.

Let's denote the events as follows:

A: The bolt was manufactured by machine A.

B: The bolt was manufactured by machine B.

C: The bolt was manufactured by machine C.

D: The bolt is defective.

We are given the following probabilities:

P(A) = 0.25 (machine A's output percentage)

P(B) = 0.35 (machine B's output percentage)

P(C) = 0.40 (machine C's output percentage)

P(D|A) = 0.05 (probability of defect given machine A)

P(D|B) = 0.04 (probability of defect given machine B)

P(D|C) = 0.02 (probability of defect given machine C)

We need to find the probability P(A|D), P(B|D), and P(C|D), which represent the probability that the bolt was manufactured by each machine given that it is defective.

According to Bayes' theorem:

P(A|D) = (P(D|A) * P(A)) / P(D)

P(B|D) = (P(D|B) * P(B)) / P(D)

P(C|D) = (P(D|C) * P(C)) / P(D)

To calculate P(D), we can use the law of total probability:

P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)

Substituting the given values, we can calculate P(D) and then use Bayes' theorem to find P(A|D), P(B|D), and P(C|D).

P(D) = (0.05 * 0.25) + (0.04 * 0.35) + (0.02 * 0.40)

= 0.0285

P(A|D) = (0.05 * 0.25) / 0.0285

≈ 0.4386

P(B|D) = (0.04 * 0.35) / 0.0285

≈ 0.4930

P(C|D) = (0.02 * 0.40) / 0.0285

≈ 0.0702

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Related Questions

If the beaters on a mixer make 800 revolutions in 5 minutes, what is the average rotational speed of the beaters

Answers

The average rotational speed of the beaters on a mixer can be calculated by dividing the total number of revolutions by the time taken.

In this case, if the beaters make 800 revolutions in 5 minutes, the average rotational speed of the beaters can be determined.

To find the average rotational speed of the beaters, we divide the total number of revolutions by the time taken. In this case, the beaters make 800 revolutions in 5 minutes. To calculate the average rotational speed, we divide 800 revolutions by 5 minutes:

Average rotational speed = 800 revolutions / 5 minutes = 160 revolutions per minute

Therefore, the average rotational speed of the beaters on the mixer is 160 revolutions per minute. This means that, on average, the beaters complete 160 full rotations within a minute. It provides an indication of the speed at which the beaters are rotating and is useful for understanding the performance and efficiency of the mixer.

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The joint probability density function of X and Y is given by f(x, y) = ху x2 + 7 2 0 < x < 1,0 < y < 2 (a) Verify that this is indeed a joint density function. (b) Compute the density function of X. (c) Find P{X > Y}. (d) Find P{Y > žix < į }. (e) Find E[X]. (f) Find E[Y]

Answers

The density function of X is fX(x) = 2x^3 + 14x, the value of P{X > Y} is 43/20. the value of P{Y > X}, is 7/6. the value of E[X] = 76/15 and E[Y] = 11.

To verify that f(x, y) is a joint density function, we need to check two conditions:

  1. The function is non-negative for all x and y.

  2. The integral of f(x, y) over the entire xy-plane is equal to 1.

Let's check these conditions:

1. Non-negativity:

  f(x, y) = xy * (x^2 + 7)

  Since x and y are both greater than 0, and (x^2 + 7) is non-negative for all x, y, it follows that f(x, y) is non-negative for all x, y.

2. Integral over the entire xy-plane:

  ∫∫ f(x, y) dA = ∫[0,1] ∫[0,2] xy * (x^2 + 7) dy dx

                 = ∫[0,1] [(1/2)xy^2 + 7y] evaluated from y=0 to y=2 dx

                 = ∫[0,1] (x + 14) dx

                 = [(1/2)x^2 + 14x] evaluated from x=0 to x=1

                 = (1/2) + 14 - 0

                 = 15/2

  Since the integral evaluates to a finite value, and this value is equal to 1 (15/2 = 1), we can conclude that f(x, y) is indeed a joint density function.

To compute the density function of X, we need to integrate the joint density function over all possible values of y:

  fX(x) = ∫[0,2] f(x, y) dy

        = ∫[0,2] xy * (x^2 + 7) dy

        = x(x^2 + 7) * ∫[0,2] y dy

        = x(x^2 + 7) * [(1/2)y^2] evaluated from y=0 to y=2

        = x(x^2 + 7) * (1/2)(2^2 - 0^2)

        = 2x(x^2 + 7)

        = 2x^3 + 14x

Therefore, the density function of X is fX(x) = 2x^3 + 14x.

To find P{X > Y}, we need to integrate the joint density function over the region where X > Y:

P{X > Y} = ∫∫[X>Y] f(x, y) dA,

where [X>Y] represents the region where X > Y.

The region where X > Y corresponds to the triangular region above the line y = x in the given range. We can set up the integral as follows:

P{X > Y} = ∫[0,1] ∫[y,1] f(x, y) dx dy

           = ∫[0,1] ∫[y,1] xy * (x^2 + 7) dx dy

The value of the double integral ∫[0,1] ∫[y,1] xy * (x^2 + 7) dx dy is 43/20.

To find P{Y > X}, we need to integrate the joint density function over the region where Y > X:

P{Y > X} = ∫∫[Y>X] f(x, y) dA,

where [Y>X] represents the region where Y > X.

The region where Y > X corresponds to the triangular region below the line y = x in the given range. the value of the double integral ∫[0,1] ∫[0,x] xy * (x^2 + 7) dy dx is 7/6.

Hence, P{Y > X} = 7/6.

To find E[X], we need to calculate the expected value of X by integrating X multiplied by its density function:

E[X] = ∫[0,1] x * fX(x) dx

= ∫[0,1] x * (2x^3 + 14x) dx

The value of the integral ∫[0,1] x * (2x^3 + 14x) dx is 76/15.

Hence, E[X] = 76/15.

To find E[Y], we need to calculate the expected value of Y by integrating Y multiplied by its density function:

E[Y] = ∫[0,2] y * fY(y) dy

= ∫[0,2] y * (∫[0,1] f(x, y) dx) dy

The value of the integral ∫[0,2] y * fY(y) dy = ∫[0,2] y * (∫[0,1] f(x, y) dx) dy is 11.

Hence, E[Y] = 11.

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An object in motion in the plane has acceleration vector a(t) = (sint, e) for 0 Sts 5. It is at rest when t = 0. What is the maximum speed it attains? 2.217 O 1.022 O 3.162 O 1.414

Answers

The maximum speed attained by the object in motion can be determined by finding the magnitude of its velocity vector. The maximum speed attained by the object is approximately 2.217.

To find the maximum speed, we need to integrate the acceleration vector to obtain the velocity vector. Integrating the x-component of the acceleration, sin(t), yields -cos(t) + Cx, where Cx is a constant of integration. Integrating the y-component of the acceleration, e, gives et + Cy, where Cy is another constant of integration.

Applying the initial condition that the object is at rest when t = 0, we find that Cx = 1 and Cy = 0. Therefore, the velocity vector is given by v(t) = (-cos(t) + 1, et).

The magnitude of the velocity vector is sqrt((-cos(t) + 1)^2 + (et)^2). To find the maximum speed, we need to determine the maximum value of this magnitude. Since the expression inside the square root is always positive, the maximum speed occurs when the expression inside the square root is maximized.

By taking the derivative of the expression inside the square root with respect to t and setting it equal to zero, we can find the critical point. Solving this equation yields t = pi/2. Substituting this value into the magnitude of the velocity, we find that the maximum speed is sqrt((1 + 1)^2 + (e(pi/2))^2) = sqrt(4 + e^2) ≈ 2.217. Therefore, the maximum speed attained by the object is approximately 2.217.

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