## Description

## 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.