Business Simulation Exam

ADM 3305 – Practice Problems Solution Problem 10

Consider a sequence of items from a production process, with each item being graded as good or defective. Suppose that a good item is followed by another good item with probability 0.99 and is followed by a defective item with probability 0.01. Similarly, a defective item is followed by another defective item with probability 0.12 and is followed by a good item with probability 0.88. Graphical representation of Xn:

a) If the first item is good, what is the probability that the first defective item to appear is the

fifth item?

Answer = p00´p00´p00´p01 = 0.993´0.01 ≈ 0.01 b) What is the probability that the fourth item is defective given that the first item is

defective?

Answer = p11´p11´p11 + p11´p10´p01 + p10´p01´p11 + p10´p00´p01 = 0.123 + 2´0.12´0.88´0.01 + 0.88´0.99´0.01

≈ 0.01 c) Let Xn denote the quality of the n-th item with Xn = 0 meaning “good” and Xn = 1 meaning

“defective”. Model Xn, the quality of the item produced, as a Markov chain by providing its corresponding graphical representation (i.e., states and transition probabilities).

See graphical representation above.

d) In the long run, what is the probability that an item produced by this process is defective?

Flow balance equations: 𝜋" = 0.99𝜋" + 0.88𝜋) (1) 𝜋) = 0.01𝜋" + 0.12𝜋) (2) 𝜋" + 𝜋) = 1 (3)

0 1 0.99 0.12

0.01

0.88

Using equations (1) and (3), we obtain: (1): 0.01𝜋" − 0.88𝜋) = 0 (3): 0.88𝜋" + 0.88𝜋) = 0.88 Then, 𝜋" =

".-- ".-.

≈ 0.99 and 𝜋) ≈ 0.01. Answer: 𝜋) ≈ 1%

e) What would be the answers to parts a) and b) based on one replication of the simulation of

the production process. Use the uniform random numbers in Problem 9.

Assuming that the first item is good, we have:

Current State U(0,1) Next State 0 0.5118 (> 0.01) 0 0 0.2836 (> 0.01) 0 0 0.8416 (> 0.01) 0 0 0.2951 (> 0.01) 0 0 0.7105 (> 0.01) 0 0 0.2756 (> 0.01) 0 0 0.1561 (> 0.01) 0 0 0.7259 (> 0.01) 0 0 0.8209 (> 0.01) 0 0 0.0857 (> 0.01) 0

Assuming that the first item is defective, we have:

Current State U(0,1) Next State 1 0.5118 (> 0.12) 0 0 0.2836 (> 0.01) 0 0 0.8416 (> 0.01) 0 0 0.2951 (> 0.01) 0 0 0.7105 (> 0.01) 0 0 0.2756 (> 0.01) 0 0 0.1561 (> 0.01) 0 0 0.7259 (> 0.01) 0 0 0.8209 (> 0.01) 0 0 0.0857 (> 0.01) 0

Answers: a) Probability = 0 b) Probability = 0