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REY [17]
3 years ago
5

A TV set contains five circuit boards of type A, five of type B, and four of type C. The probability of failing in its first 500

0 hours of use is 0.03 for a type A circuit board, 0.04 for a type B circuit board, and 0.03 for a type C circuit board. Assuming that the failures of the various circuit boards are independent of one another, compute the probability that no circuit board fails in the first 5000 hours of use.
Mathematics
1 answer:
nekit [7.7K]3 years ago
8 0

Answer:

0.903264

Step-by-step explanation:

Given that a TV set contains five circuit boards of type A, five of type B, and four of type C. The probability of failing in its first 5000 hours of use is 0.03 for a type A circuit board, 0.04 for a type B circuit board, and 0.03 for a type C circuit board.

Let A' = the event that A fails, B' =  B fails and C'= C fails.

Probability that no circuit board fails = Prob (A'B'C')

= P(A')P(B')P(C')  (since A,B, C are independent, their complements are independent

= (1-0.03)(1-0.04)(1-0.03)

= 0.903264

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A grinding wheel manufacturer designed a new grinding wheel. Repeated tests were conducted on wheels of approximately the same w
sergejj [24]

Answer:

a) a=225 +0.674*16.5=236.121

So the value of height that separates the bottom 75% of data from the top 25% is 236.121.  

b) P(X \geq 3) = 1-P(X

c) P(\bar X \geq 225)=1- P(\bar X

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Part a

Let X the random variable that represent the cuts of a population, and for this case we know the distribution for X is given by:

X \sim N(225,16.5)  

Where \mu=225 and \sigma=16.5

For this part we want to find a value a, such that we satisfy this condition:

P(X>a)=0.25   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.75 of the area on the left and 0.25 of the area on the right it's z=0.674. On this case P(Z<0.674)=0.75 and P(z>0.674)=0.25

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=0.674=\frac{a-225}{16.5}

And if we solve for a we got

a=225 +0.674*16.5=236.121

So the value of height that separates the bottom 75% of data from the top 25% is 236.121.  

Part b

For this case we know that the individual probability of select one wheel with a cutting rate higher than the calculated value in part a is 0.25, and we select n =10 so then we can use the binomial distribution for this case:

X\sim Bin(n=10, p=0.25)

And we want this probability:

P(X \geq 3) = 1-P(X

We can find the individual probabilities like this:

P(X=0)=(10C0)(0.25)^0 (1-0.25)^{10-0}=0.0563

P(X=1)=(10C1)(0.25)^1 (1-0.25)^{10-1}=0.1877

P(X=2)=(10C2)(0.25)^2 (1-0.25)^{10-2}=0.2816

P(X \geq 3) = 1-P(X

Part c

For this case we know that the distribution for the sample mean is given by:

\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})

And we want this probability:

P(\bar X \geq 225)

And for this case we can use the complement rule and the z score given by:

z= \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}

And if we replace we got:

P(\bar X \geq 225)=1- P(\bar X

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3 years ago
Find the point, M, that divides segment AB into a ratio of 2:3 if A is at (0.15) and B is at (20.0)
timama [110]

Answer:

M = (8,9)

Step-by-step explanation:

Notice that the points (0,15), and (20,0) form with the origin of coordinates (0,0),  a right angle triangle (please see attached image). This triangle has twon perpendicular sides of length 15 and 20 respectively. Therefore, we can find the length of the segment that joins points A (0,15) and B (20,0) by finding the length of the hypotenuse in a right angle triangle (with the Pythagorean Theorem):

AB=\sqrt{15^2+20^2} =\sqrt{625} =25

Now, to get a 2:3 proportion on Segment AB which is of length 25, we need to divide it in five equal parts (see the picture on the right of the attached image), and place point M at two of these divisions from point A (0,15) and along segment AB.

In order to find the appropriate location in (x,y) coordinates, we consider a smaller triangle (pictured in orange in the image) that is similar to the first larger triangle (pictured in blue). Notice that if the length of AB is 25,  each of its five equal divisions would be of length "5", and therefore two of them will render a length of "10" (which is the hypotenuse of this smaller right angle triangle.

Now, in order to find the sides of this smaller triangle (which can give us the clues on the horizontal and vertical coordinates of point M), we can use proportions.

To find the length "x" of the horizontal side , we do:

\frac{x}{10} =\frac{20}{25} \\x=\frac{10*20}{25} \\x=8

To find the length "y" of the vertical side , we do:

\frac{y}{10} =\frac{15}{25} \\y=\frac{10*15}{25} \\y=6

Then, the coordinate "x" of point M will be "8", while we can calculate the y position of point M subtracting "6" from 15 (the length of the vertical side in the original triangle). This gives us the coordinates (8,9) for point M as marked in orange in the picture.

7 0
3 years ago
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