Question

Have to show work for every problem

4. A company uses three plants to produce a new computer chip. Plant A produces 30% of the chips. Plant B produces 45% of the chips. The rest of the chips are produced by plant C. Each plant has its own defectiv rate. These are: plant A produces 3% defective chips, plant B produces 1% defective chips, plant C produces 5% defective chips. Hint: draw a tree diagram. (a) Construct a tree diagram and write the appropriate probability on each of the branches. (b) Find the probability that a randomly chosen chip is defective and from plant B. Answer: 0.0045 (c) Find the probability that a randomly chosen chip from Plant C is defective. Answer: 0.05 (d) Find the probability that a randomly chosen chip is defective. Answer: 0.026 (e) Find the probability that a randomly chosen defective chip is from plant C. (Hint: look at Example 12 on Pages 3-22 to 3-24.) Answer: 0.481 (f) Find the probability that a randomly chosen chip is not from plant A. Answer: 0.700 (g) Find the probability that a randomly chosen defective chip is not from plant A 0.654 (3.P.2) 5 . Answer:

Answer 1

Let P=Probability

Let Defective =D and non-defective =D^c

4.

(a)

The probability tree diagram:

(b)

P(D and B) =P(BD) =P(B).P(D/B)=0.45(0.01)=0.0045​​​​​​

(c)

P(D/C) =0.05​​​​​​

(d)

P(D) =P(AD)+P(BD)+P(CD) =0.009+0.0045+0.0125 =0.026​​​​​

(e)

BY Baye's rule, P(C/D) =P(D/C).P(C)/P(D) =0.05(0.25)/0.026 =0.481

(f)

P(A^c) =1 - P(A) =1 - 0.30 =0.70​​​​​​

(g)

P(A^c/D) =1 - P(A/D) =1 - [P(D/A).P(A)/P(D)] =1-[0.03(0.30)/0.026] = 1-0.346 =0.654

5.

Let P=Probability

Let answer known =K. So, P(K) =0.70 and also P(KC) =0.70 because when the answer is known, it becomes correct answer only. So, the probability of knowing the answer and correct answer =0.70

Let answer is not known =K^c. So, P(K^c​​​​​​) =1 - 0.70 =0.30

Let correct answer =C

The probability of correct answer when it is unknown =P(C/K^c) =1/4 =0.25

Now, the probability of a randomly chosen question becoming a correct answer =P(C) =P(KC)+P(K^cC) =0.70+P(K^c).P(C/K^c) =0.70+[0.30(0.25)] =0.70+0.075 =0.775