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BartSMP [9]
3 years ago
11

A manufacturing process produces integrated circuit chips. Over the long run, the fraction of bad chips produced by the process

is 20%. Thoroughly testing a chip to determine whether it is good or bad is rather expensive, so a cheaper test is used. All good chips will past the test, but so will 7.5% of the bad chips. (a) Given that a chip passes the test, what is the probability that it is a good chip? Use at least 3 decimal places. (b) If the company sells all chips that pass the cheaper test, what percentage of sold chips will be bad? Use at least 3 decimal places.
Business
1 answer:
zysi [14]3 years ago
7 0

Answer:

(a) Given that a chip passes the test, what is the probability that it is a good chip?

LetB = {the chip is good}

A={the chip passes the cheap test}.

Bc={the chip is bad}

Ac={the chip fails the cheap test}

P(A | B) = 1

P(A | B c ) = 0.075

= \frac{P(A | B)P(B)}{P(A | B)P(B) + P(A | Bc)P(Bc)} = \frac{1.0.8}{1.0.8+ 0.075 · 0.2}  ≈ 0.9751

(b) If the company sells all chips that pass the cheaper test, what percentage of sold chips will be bad?

P(B c  |A) =  1 − P(B | A) = 1 -  0.9751 = 0.0249

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The amount Teresa will have accumulated when this certificate matures is $2,452.16.

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Note: This question is not complete as some important data are omitted. The complete question is therefore provided before answering the question as follows:

At the end of every 3 months, Rita deposits $100 into an account that pays 5% compounded quarterly. After 5 years, she puts the accumulated amount into a certificate of deposit paying 8.5% compounded semiannually for 1 year. When this certificate matures, how much will Teresa have accumulated?

The explanation of the answers is now provided as follows:

Step 1: Calculation of accumulated amount after 5 years.

Since the deposits are paid at the end of every 3 months, the accumulated amount after 5 years can be calculated using the formula for calculating the Future Value (FV) of an Ordinary Annuity as follows:

FV5 = P * (((1 + r1)^n1 - 1) / r) ................................. (1)

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FV5 = Future value or accumulated amount after 5 years = ?

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r = Quarterly interest rate on the account = Interest rate on the account / Number of quarters in a year = 5% / 4 = 0.05 / 4 = 0.0125

n = number of quarters = 5 years * Number of quarters in a year = 5 * 4 = 20

Substituting the values into equation (1), we have:

FV5 = $100 * (((1 + 0.0125)^20 - 1) / 0.0125) =  $2,256.30

Therefore, the accumulated amount after 5 years is $2,256.30.

Step 2: Calculation of the amount Teresa will have accumulated when this certificate matures.

This can be calculated using the simple future value (FV) as follows:

FVM = FV5 * (1 + R)^N ……………………… (2)

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N = number of semiannuals = 1 year * Number of semiannuals in a year = 1* 2 = 2

Substituting the values into equation (2), we have:

FVM = $2,256.30 * (1 + 0.0425)^2 = $2,452.16

Therefore, the amount Teresa will have accumulated when this certificate matures is $2,452.16.

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