In the manufacturing of a product, 88% of the products made are not defective. O

ID: 3221274 • Letter: I

Question

In the manufacturing of a product, 88% of the products made are not defective. Of those inspected, 7% of the good ones are rules as defective and not shipped. Only 2% of the defective products are approved and shipped. If a product is shipped, what is the probability that it is defective? Suppose that you are on a game show, and you're given the choice of three doors. Behind one floor is a new car, and behind each of the other doors is a cow. You pick a door (say. you pick door #1), but rather than open the door, the host opens one of the remaining doors (say, floor #2) which reveals a cow (he knows what's behind each door). He then offers you a choice: you can stay with door #1, or switch to door #3. Is it to your advantage to switch to door #3? Prove that a graph with at least two vertices has a pair of distinct vertices with the same degree.

Explanation / Answer

Question 5:

Let non defective be represented by the notation ND and defective be represented as D.

Let the test giving the defective notation be represented as TD that is tested defective and TND be tested non defective. Then we are given that:

P(ND) = 0.88 . Therefore P(D) = 1 - P(ND) = 0.12

P( TD | ND ) = 0.07 because 7% of the good ones are ruled as defective. Therefore we get: P(TND | ND) = 0.93

P( TND | D) = 0.02 because only 2% of the defective products are tested OK and shipped.

Now we have to find that if the product is tested OK, then what is the probability that is is defective. This would be:

P ( D | TND ) = ?

From Bayes thorem we get:

P ( D | TND )P(TND) = P(TND | D) P(D)

Now we will compute P(TND) using the law of total probability as:

P(TND) = P(TND | D) P(D) + P(TND | ND) P(ND) = 0.02*0.12 + 0.93*0.88 = 0.0024 + 0.8184 = 0.8208

Therefore now putting all the values into the Bayes equation we get:

P ( D | TND )P(TND) = P(TND | D) P(D)

P ( D | TND )*0.8208 = 0.02*0.12

P ( D | TND ) = 0.0024 / 0.8208 = 0.0029

Therefore 0.0029 is the required probability here.

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In the manufacturing of a product, 88% of the products made are not defective. O

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