# Homework 1 ISyE 6420 Circuit solution

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## 1. Circuit.

A circuit S consisting of six independent elements E1, . . . , E6 is connected as
E1
E5
E6
E3
E2
E4
E7

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 for
p = 0.4.

(b) If the machine failed, what is the probability that the component which fails with
probability 1/2 actually failed.