Problem

A function $g(x)$ is convex if the chord connecti…

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Problem 165 Easy Difficulty

Exercise 49 introduced two machines that produce wine corks, the first one having a normal diameter distribution with mean value 3 $\mathrm{cm}$ and standard deviation. 1 $\mathrm{cm}$ and the second having a normal diameter distribution with mean value 3.04 $\mathrm{cm}$ and standard deviation .02 cm. Acceptable corks have diameters between 2.9 and 3.1 $\mathrm{cm} .$ If 60$\%$ of all corks used come from the first machine and a randomly selected cork is found to be acceptable, what is the probability that it was produced by the first machine?

Answer

.5062

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Intro Stats / AP Statistics

Probability with Applications in Engineering, Science, and Technology

Chapter 3

Continuous Random Variables and Probability Distributions

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Video Transcript

Okay, So the mean and standard deviation first machine sorry is, um is equal to three on us is equal to zero. But one perspective for the second machine, you know, im is equal to 3.4. Under standard deviation A Z equals 0.2 Note that I could be using a signal to represent sound variations. Well, so now what? We're gonna do this Set variables letting X be acceptable courts except folks on Let's let he won a prison. The event that a machine, that court comes out from the first machine. So first machine, right. And okay, so now we're given that P if you want. Is he cool? Did you put six on D P of two? Which is just a probability that comes out of the second machine is a job for which is just one, but it's your six. Okay, So now the range of acceptable corks were given is between 2.9. Andi, do you think? Yeah. Yeah. Okay. So now I have to find the probability that screws from the first machine are accepted. Just a probability to nine minus three. Or is super sperm one less than X minus mu percent of creation. Just playing in the general for Do you know one three over ship that is equal to p native. One was lesson. See, Weston. So therefore the probability she Sorry one manage the probability G is less than or equal to negative one. Now, just plugging in those values you get 0.8 four 13 minus 0.1587 which gives us a final answers your 0.68 27 Okay, now the next step has to find the probability except works from second machine. So again, we're gonna use the same formula. Thank you. Our new values. So nice to print your four over Is Europe punch. You too is less than X minus mu over. Signal to the standard deviation. Just listen to the 103 puts here. Four. I love her. Ju wasn't you too? So then, that is just the probability C is greater and negative. Seven less than three. Which will give us your 0.99 seven if you follow this thing. Methodology. Okay. So now let's let very well a represent the event. The selected cork is found to be acceptable. Okay, Except tell me probably the selected cork is acceptable, but it to machines is just the probability of A Which is equal does your 0.6, which is the probability off the court coming out of the first machine times the probability that the work is acceptable? 0.68 to 7. Plus you look for it's a probability of cooker for a second machine, times the probability of their being. It's absolutely 9987 and that will give us a final answer. Was your 0.8 year all right?

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