2. A factory producing headsets has three assembly lines labeled A, B and C. Lin
ID: 3040834 • Letter: 2
Question
2. A factory producing headsets has three assembly lines labeled A, B and C. Line A produces 20% of the headsets, line B produces 30% of the headsets and line C produces 50% of the headsets. Quality assurance studies indicate that 2% of the headsets produced in line A are defective, l % ofthe headsets produced in line B are defective and 3% of the headsets produced in line C are defective. A headset is randomly selected from the output of the factory. (a) What is the probability that it is defective? Given that it is defective, what is the probability that it came from line B? 3. A binary erasure channel is a communication channel model in a transmitter sends a bit and the receiver either receives the bit or it receives a message that the bit was not received (i.e., it was "erased"). Specifically, when a random variable X, which is an alphabet 0 or 1, is transmitted the received variable Y is an alphabet that is 0, 1 or e, where e is the erasure symbol that signifies that the transmitted symbol was not received. We consider an asymmetric binary erasure channel as follows. There are two input symbols X1 and X2, and three output symbols ,, ½ and E. When X1 is transmitted, Y1 is received with probability 0.9 and E is received with probability 0.1. Similarly, when X2 is transmitted, Y2 is received with probability 0.8 and E is received with probability 0.2. The diagram for this scheme is shown in the figure below where it is assumed that the probability that X1 is transmitted is 0.6 and the probability that X2 is transmitted is 0.4; that is, P[X] = 0.6, P(X2-0.4. 4 (a) What is the probability that the erasure symbol E is received? (b) Given that the erasure symbol was received, what is the probability that X2 was transmitted?Explanation / Answer
(2)
Let's assume that 1000 sets are produced overall.
According to the data given we have:
Sets produced by A = 200
Sets produced by B = 300
Sets produced by C = 500
Defective sets amongst those produced by A = 4
Defective sets amongst those produced by B = 3
Defective sets amongst those produced by C = 15
(a)
Probability that the randomly chosen headset is defective = Total defective/Total produced = (4+3+15)/1000 = 0.022
(b)
Probability that the randomly chosen headset is from B, given that it is defective = Total defective sets produced by B/Total defective = 3/(4+3+15) = 0.136
Hope this helps !
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