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A company makes electric motors. The probability an electric motor is defective is 0.01.So, out of 300 electric motors, the expected number of defective motors is 3...

Question

A company makes electric motors. The probability an electric motor is defective is 0.01.So, out of 300 electric motors, the expected number of defective motors is 300 X 0.01 3. With this in mind, use a Poisson distribution to estimate the probability that exactly 5 motors are defective: (Use at least 5 decimal places)Notice that; if we are assuming that the defectiveness of each motor is independent from the others, then the number of defective motors out of the 300 should follow a Binomial dist

A company makes electric motors. The probability an electric motor is defective is 0.01. So, out of 300 electric motors, the expected number of defective motors is 300 X 0.01 3. With this in mind, use a Poisson distribution to estimate the probability that exactly 5 motors are defective: (Use at least 5 decimal places) Notice that; if we are assuming that the defectiveness of each motor is independent from the others, then the number of defective motors out of the 300 should follow a Binomial distribution with n 300 and p 0.01 Use this distribution to calculate the probability that exactly 5 motors are defective, and compare it with your estimate from part (a). (Again; calculate the answer to at least 5 decimal places_ (In the case where n is very large and p is very small in a Binomial distribution, the Poisson distribution is computationally easier to handle but gives good , approximations for the probabilities. This is essentially what s known as the "law of rare events"



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Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.
An auto parts seller finds that 1 in every 100 parts sold is defective. Find the probability that (a) the first defective part is the tenth part sold, (b) the first defective part is the first, second, or third part sold, and (c) none of the first 10 parts sold are defective.

27 uh, probability or four is ableto 6000 C four times we'll find 04 hour, four times one minus 4.4 for our 6000 minus minus four, which is all 40.1 to 5. Could shouldn't be switching language and be, You know that the mean is equal to end times B, which is 6000 times 4.4 which is 2.4 eso. The probability of X equal to four is different to 2.4 if our negative 2.4 over for factorial, which is a 0.125 or zero a.

So we know that the probability that uh someone is in favor of using nuclear energy for electricity is 0.57 or 57%. And we want to determine if eight people are chosen at random. What is the likelihood that four of those people will end up saying that they are in favor of using nuclear or energy? And this is the binomial setting. So we can use it told us we can use our technology. So why don't we use it? And this will be a binomial P. D. F. And we're going to plug in the eight for the number of trials. We're going to have the probability of success 3.57 and then we'll have four for our number of our our X. Value. And when you hit the distribution button, if you're on an 84 it almost is easier to go upward rather than downward. But anyway our trials are eight R. P value is 80.57 for for the X. Value. And we end up finding that probability comes out to be a 0.25 to six Part B. We want to find the probability that less than five people out of the eight would be in favour, which means we're really finding X being less than or equal to four. So that 01234 And that is going to be most easily found by by using our binomial setia and having eight 0.57 and four. And that will accumulate all of those probabilities very slick. And let me tuck that in. So eight for my trials, my 80.57 from my probability of success. My ex value is four. And again that is make sure you're on the CDF, not the pdf. And that naturally is going to be a bigger answer than the previous because this answer was included in this answer and I've rounded these off a bit and then the last one we want to find the probability that the number of people is at least three which is the complement of this includes three all the way up to eight. And we want the compliment which will be 01 and two. Yeah two. So if we do one minus are binomial syria mhm of 8.57 and two be a compliment. Well actually find our answer for this scenario and we want to change that value to A two and when we do that we find out that that is 0.9289 well rounded to three. Mhm. So all three of these were a binomial setting.

The probability of exactly three engines functioning properly echoes the term data can change .95 Cubed and hands. The coefficient is for Times points 95 Cube Times .05, and the answer is .171.

The number of surface flaws in plastic panels used in the interior of automobiles has a Poisson distribution. So because it has a Poisson distribution, we're going to be able to find our probabilities utilizing the formula P of X equals E to the negative lambda times lambda to the X. Power all over X. Factorial. And it goes on to read that the mean is .05 flaws per square foot. So the mean is .05 flaws In one square foot. Assume that an automobile interior contains 10 square feet. So what we need to do is we've got to determine our λ based on 10 square feet. So we're going to use a proportion .05 flaws In one square foot would be comparable to how much In 10 sq ft. And if we were to cross multiply and solve for lambda, we will have a lambda value of five flaws In that 10 square foot space. So part A is asking you what is the probability that are, there are no surface flaws in the autos interior. So what we're going to say is X is going to represent the number of surface flaws in the interior And keep in mind that interior is 10 square feet. So when we are trying to determine the probability, we are trying to determine the probability that there are none. So we're going to say we want the probability when X is zero. So we'll substitute our values into the formula. So we'll have E to the negative lambda times lambda raised to the X. Power all over X. Factorial. Now using your knowledge of algebra anything to the zero power is one and zero factorial is one. So really this is just going to be E to the negative 10.5 power. And when you calculate that you are going to get a value of about 0.606530 6597. And based on the recommendations of your professor or your teacher, you might round it to be .6065 part B in part B. We are now talking about 10 cars being sold to a rental company And we want to know what is the probability that none of the 10 cars has any surface for us. So we're gonna have to change our variable so we're gonna change our variable to why being the number of cars with surface flaws. And if you think about the fact that the probability of no surface flaws is the answer to part A 606,530 6597. Than the probability of having surface flaws, Whether it be one or many will be that complement. So it's gonna be one minus the 10.606 or 0.393 4693403. So when we are doing the probability that none of the 10 cars then we're saying what's the probability that y equals zero. Now in this instance we are now talking in terms of a binomial probability, we're no longer talking in terms of a person probability because we're asking in car number one doesn't have flaws or no car number two doesn't have flaws or no. So we are recording how many cars have flaws. So when it comes time to do binomial probability we are going to use the formula P of Y equals n C. Y times the probability of success which is P to the Y power times one minus P to the n minus Y power. So P is the probability of success and we are talking about having flaws so therefore r p value Is going to be the .3934693403. So the probability that the no cars have a flaw is going to be 10 because there were 10 cars C zero. The probability of having surface flaws was .3934693403 raised to the zero power. And then we're multiplying that by 1 -4. Which would be this value .6065306597 to the 10 0 Power or to the 10th Power. So the probability of no cars with flaws when we sell them to the rental company will be .006737947. Right And again based on your professor or teacher's recommendation You might round it and I usually go to four decimal places. So .0067. And then for part see We want if 10 cars are sold to the rental company, What's the probability that at most one has surface flaws? So we want probability at most one. So we're talking cars. So why is less than or equal to one? Which is the same thing as saying the probability of why equaling zero plus? The probability of why equaling one. We just found the probability of why being zero to be this value. So we could say .006737947. And then for the probability of one we're going to utilize our binomial formula which is right here. But this time where Y is equal to one so we'll have plus 10. See one multiplied by the probability Of .3934693403 to the first power Multiplied by one -P. Which is going to be .606) 530 6597 To the N -1 power which would be to the 9th power. So this will end up being .006737947. Plus when you calculate all of this out you are going to get a value of .0437 10 4954. And when you add those two probabilities together we are getting an overall probability of .050448 44, 2 4. And again based on your professors recommendation, I round to four decimal places. So I would say .0504. So in summary, the probability of a car with One single car with no interior surface flaws was zero Or 6065. The probability of Selling 10 cars to the rental company and no cars Of the 10 having an interior flaw is .0067. And the probability of that most, one of those 10 cars having an interior flaw would be .0504.


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