Question 4 [5 points] Cement Co. has 40 cement trucks and of...

Question 4 [5 points] Cement Co. has 40 cement trucks and of these 40 trucks 30 have engine defects. A Sample of 4 trucks is randomly selected. What is the probability that exactly 3 of those tested have engine defects? For full marks your answer should be accurate to at least four decimal places. P(X=3) = 0 Question 5 [6 points] A company uses large panels of sheet metal in the manufacturing of tandem trailers. There is an average of 3 blemishes per panel. Calculate the probabilities if the blemishes follow a Poisson process. For full marks your answer should be accurate to at least four decimal places. a)Exactly 3 blemishes on 2 panels? P(x) = 0 b)Fewer than 3 blemishes on 2 panels? P(x) = 0 c)Either 3 or 4 blemishes on 3 panels? P(x) = 0

Transcript

00:01 We're told in a trucking company there are 40 trucks, and of those 40 trucks, 30 have defects.

00:07 And so they're going to take a random sample of, they're going to sample four, and we want to know the probability of exactly three, x is three, of those four having defects.

00:25 And so this is a hypergeometric distribution we're going to use, hypergeometric.

00:33 Metric.

00:36 And when i first did these kind of problems in my sas scores, i was thinking it was more binomial.

00:44 Like you take, oh, 30 out of 40, which is probably 0 .75.

00:49 But that doesn't work because binomial has the probability being the same for each selection.

00:53 But if there's 40 trucks and you sample four of them, the probabilities go down of them having a defect depending on as each one you select so hyper geometric accounts for that changing probability so the way this works is we have the probability this say x choose little x here x maybe star is equal to k choose little x here and then up here we've got and big n minus k i'll tell you what all these me in just a second.

01:37 And then we've got little n minus x naught all over big n choose little n.

01:55 So let's see what this is.

01:57 So ignore this little n here.

01:58 Big n here is the total of the sample.

02:01 So in this case, 40 is our big n.

02:03 K is the number, big k is the total number of defects.

02:06 There's 30 defects.

02:09 So let's put this together.

02:10 We've got big k is 30.

02:13 X naught, this is going to be 3.

02:18 This is x -naught right here.

02:20 And then n minus k.

02:23 Big n we said is 40.

02:25 Big k is 30.

02:26 So 40 minus 30 is 10.

02:29 And then we've got little n, which was the sample size.

02:33 Little n is the sample size.

02:35 So this gets little n is 4, sample size.

02:37 So it's 4 minus x -naught is 3.

02:41 So that's just going to be 1.

02:42 And then we divide that by big n which is 40, choose little n which is 4, and there we go.

02:50 This is our problem.

02:52 And then the reason for this is because, or the way i kind of reason through this is that this is the number of defects, combinations of getting 3 defects out of the 30 defective ones.

03:07 10 choose 1, this is out of the 10 ones that don't have defects, we want one that does not have a defect and then this down here for each is for this is the combinations of two of the forty we want four of the forty any doesn't really matter but whether they have defects just the number of ways we can get four out of forty and then just as a reminder this formula here in general do this in in a separate color.

03:36 We'll do this.

03:38 And choose, say, x.

03:41 This is the combinations formula you get by doing n factorial divided by x factorial times n minus x quantity factorial.

03:50 Alright, so let's go ahead and get those values and get our probability.

03:55 30 choose 3 is 40, 60.

03:56 4 ,060...

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