Description

ISyE 6420
1. Circuit. A circuit S consisting of six independent elements E1,…,E6 is connected as

Figure 1: Circuit S with six independent elements
in Figure 1. The elements are operational during time interval T with probabilities
E1 E2 E3 E4 E5 E6 E7
Probability of working (p) 0.5 0.7 0.3 0.4 0.9 0.5 0.7
(a) Find the probability that the circuit is operational during time interval T.
(b) If the circuit was found operational at the time T, what is the probability that the element E6 was operational.
2. Two Batches. There are two batches of the same product. In one batch all products are conforming. The other batch contains 10% non-conforming products. A batch is selected at random and one randomly selected product from that batch is inspected. The inspected product was found conforming and was returned back to its batch.
What is the probability that the second product, randomly selected from the same batch, is found non-conforming?
Hint. This problem uses both Bayes’ rule and Total Probability. The two hypotheses concern the type of batch. For the first draw the hypotheses are equally likely (the batch is selected at random), but for the second draw, the probabilities of hypotheses are updated by the information on the result of the first draw via Bayes rule. Updated probabilities of hypotheses are then used in the Total Probability Formula for the second draw.
3. Machine. A machine has four independent components, three of which fail with probability q = 1 − p, and one with probability 1/2. The machine is operational as long as at least two components are working.
(a) What is the probability that the machine will fail? Evaluate this probability forp = 0.4.
(b) If the machine failed, what is the probability that the component which fails withprobability 1/2 actually failed.
2