An average joe is working for a manufacturing business has a...

An Average Joe is working for a manufacturing business has a 98% chance of correctly identifying defective items and a 1% chance of incorrectly classifying a good item as defective. The business has evidence that its line produces defective items 0.8% of the time. 1) What is the probability that an item selected for inspection is classified as defective? 2) If an item selected at random is classified as non-defective, what is the probability that it is indeed acceptable?

Transcript

00:01 All right, we're told that there's someone working on an assembly line where they're classifying products as either defective or non -defective.

00:08 And so i've defined the event cd as the event that the product is classified as defective.

00:15 Event c -n, the probability, or sorry, the event that the product is classified as non -defective or good.

00:23 The event d, the probability or the event that the product is actually defective.

00:27 And the event n is the event that the product is actually non -defective or good.

00:34 And so we're told that our worker can correctly identify defective products with a rate or 98 % of the time, meaning that the conditional probability that it is classified defective, given that it is defective, is equal to 0 .98.

00:50 Similarly, we're told that he mistakenly identifies non -defective products as defective 1 % of the time.

00:58 So the conditional probability that a product has classified as defective, given that it's not defective, is equal to 0 .01.

01:06 And finally, we're told that the assembly line produces defective products at a rate of 0 .8%.

01:13 So the probability of having a defective item is 0 .008.

01:16 So first, we're interested in the probability that a product is classified as defective.

01:22 And because we know that products can either be defective or non -defective, the probability that they're classified as defective is equal to the probability that they're classified as defective and that they're defective plus the probability that they're classified as defective and that they're good or non -defective.

01:41 And so we can use our conditional probability notation that the probability that an item is defective times the probability that it's classified as defective, given that's defective, is equal to our probability that it is classified as defective and it is defective to get that this value is equal to 0 .008 times 0 .98, which gives us probability 0 .00784.

02:19 Similarly, we can do this for the non -defective case, the probability that an item is classified as defective, but it is not actually defective is equal to the probability that an item is not defective, which is 1 minus 0 .008, or 0 .992 times the probability that it is classified as defective given that it's not, which is 0 .01, which is equal to 0 .00992.

02:54 So our probability of being classified as defective is going to be equal to 0 .00784 plus 0 .00992, which is equal to 0 .01776.

03:18 So approximately 1 .8 % of all products are classified as defective.

03:28 Then we're asked what is a conditional probability that given a product is classified as not defective or good, what is the probability that it actually is good? or this is equal to the probability that a product is classified as non -defective and that it is non -defective divided by the probability that it is not defective...

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