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Points) second-stage smog alert was called in a certain area of Los Angeles County in which there are 40 industrial firms. An inspector will visit 10 randomly selec...

Question

Points) second-stage smog alert was called in a certain area of Los Angeles County in which there are 40 industrial firms. An inspector will visit 10 randomly selected firms to check for violation of regulations. 15 firms are actually violating at least one - regulation. (4 points) Find the probability that exactly 5 of the 10 visited firms are in violation of at least one regulation: (3 points) What is the expected number of firms (among the 10 visited by the inspector) that violate at least on

points) second-stage smog alert was called in a certain area of Los Angeles County in which there are 40 industrial firms. An inspector will visit 10 randomly selected firms to check for violation of regulations. 15 firms are actually violating at least one - regulation. (4 points) Find the probability that exactly 5 of the 10 visited firms are in violation of at least one regulation: (3 points) What is the expected number of firms (among the 10 visited by the inspector) that violate at least one regulation? 6Fage



Answers

A second-stage smog alert has been called in an area of Los Angeles County in which there are 50 industrial firms. An inspector will visit 10 randomly selected firms to check for violations of regulations.
(a) If 15 of the firms are actually violating at least one regulation, what is the pmf of the number of firms visited by the inspector that are in violation of at least one regulation?
(b) If there are 500 firms in the area, of which 150 are in violation, approximate the pmf of part (a) by a simpler pmf.
(c) For $X=$ the number among the 10 visited that are in violation, compute $E(X)$ and $\operatorname{Var}(X)$ both for the exact pmf and the approximating pmf in part (b).

Yeah this problem we are told that we have 20 firms under suspicion. Mhm. three of them are in violation. And so that means we're three or under violation and 17 or not. Now in a we want to find the probability that the inspection of five of the firms will find no violation. Now what this means is that we're choosing five of them but we want no violations. And so from the 17 that have not violated anything. We need to choose five of them. But then we're going to put this over 225 which is that's the total number of different possibilities that we can choose. Five From our 20 different options. And so now we evaluate 17-5 over 25 Which is .399 one. Okay. Mhm. Okay. I won't be. We want to find the probability that the plan above will find two violations. Yeah. And so choosing five of them. This means from the three violators, we're going to have two of those. which means from the 17 non violators, we'll have to have three of them because we are choosing five total, which is why our denominator is still 20 jews five because from the 20 when you're still choosing five and so now we evaluate these combinations and when we do this gives us okay, .1316.

So we know that the probability of the first person mrs Something in an inspection this .1 and then we're told the probability that the second one mrs given that the first person missed Is equal to five out of town or .5. And we want the probability that the first person mrs and Or intersected with the 2nd person mrs. And we know that we find the probability of the first missing Times the probability of the second missing, given that the first person already missed, we can't just multiply two probabilities together has to be a conditional. And lucky for us, we know the conditional. So it's just .1 times 0.5. And remember this is not the probability that the second person mrs it's a conditional probability and that's exactly what we need here. So the answer is point oh five. Yeah. Right.

This problem. We've been having the following probability density function somebody likes fine if he's in trouble now, in part a we want to ensure that this is in fact a valid probability density function now, in order for it to be valid, that means that the integral over all of the possible Y values and here are told the possible why matters go from 0 to 1 of our probability density function needs to be equal to one. So we're going to evaluate the central and then make sure that it is. In fact, it would have wanted. Now let me take our anti derivative here. This is negative one minus Y to the fifth power, evaluated for more equal 0 to 1. So this is negative zero to the fifth minus and you have one to the fifth, which is one. So we showed that it is in fact valid probability density function Yeah, on B. We want to find the probability that a company spends less than 10% of its budget on environmental and pollution controls now 10% the same 100.1. And so this means we want the probability that acts is less than or equal to 0.1. This will be the integral from 0 to 0.1 of our probability density function. And so once again this is negative one minus Y to the fifth, evaluated from y is zero to y 0.1. So this is negative one minus 0.10 point nine to the fifth minus one minus negative one plus one to the fifth power. Now for the next problem, I do want you to remember this anti derivative here because all I will do is change the balance on it. But then now we evaluate this so it's negative 0.9 to the fifth power past one of the fifth. And when we evaluate this should give us 0.40 95 point Yeah. On C. We want to find the probability that spends more than 50% of its budget on so on the one with the probability that that is greater than or equal to 0.5. And so I'm just going to change my balance here. We have negative one minus y to the fifth evaluated. Now we go from why is 05 to 1? It's also been negative, one minus one is zero to negative zero to fifth minus negative 0.5 to the fifth power. And so again we just evaluate this. This is going to give us 0.3125

Mhm. We want to find the probability that 10 employers must be tested in order to find three positives. And the political rate is point for. We know that the number of employees there need to be tested in order to find three positives follows a negative phenomenal distribution. With parameters are equals three mm p equals 0.4. And has the probability that 10 employers need to be tested because Nitro is two times Point of four, Cubed Times Point of 6 to A 7th. This is just a formula for the negative binomial distribution And its approximate value is .06 four or five. Also, we can think of this event as the product out to independent events. First we test nine employers and we find to poverty and this follows by a nominal distribution. And it's probability as Night shows two times .4 Squared times .6- seven. And then for the 10th test that resulted positive and Their probability is just at the .4. So this will give the same result, has about


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