A recording company obtains the blank CDs used to produce its labels from three compact disk manufacturers: I, II, and III. The quality control department of the company has determined that 3% of the compact disks produced by manufacturer I are defective, 5% of those produced by manufacturer II are defective, and 5% of those produced by manufacturer III are defective. Manufacturers I, II, and III supply 36%, 54%, and 10%, respectively, of the compact disks used by the company. What is the probability that a randomly selected label produced by the company will contain a defective compact disk?

Probability that a randomly selected label produced by the company will contain a defective compact disk is 0.0428 or 4.28%.

Step-by-step explanation:

We are given that the quality control department of the company has determined that 3% of the compact disks produced by manufacturer I are defective, 5% of those produced by manufacturer II are defective, and 5% of those produced by manufacturer III are defective. Manufacturers I, II, and III supply 36%, 54%, and 10%, respectively, of the compact disks used by the company.

Let the Probability that Manufacturer I supply compact disks = P(M I) = 0.36

Probability that Manufacturer II supply compact disks = P(M II) = 0.54

Probability that Manufacturer III supply compact disks = P(M III) = 0.10

Also, let D = defective compact disks

Probability that compact disks produced by manufacturer I are defective = P(D/M I) = 0.03

Probability that compact disks produced by manufacturer I are defective = P(D/M II) = 0.05

Probability that compact disks produced by manufacturer I are defective = P(D/M III) = 0.05

Now, probability that a randomly selected label produced by the company will contain a defective compact disk is given by;

    =  P(M I) P(D/M I) + P(M II) P(D/M II) + P(M III) P(D/M III)

    =  0.36 0.03 + 0.54 0.05 + 0.10 0.05

    =  0.0108 + 0.027 + 0.005

    =  0.0428 or 4.28 %

Hence, probability that a randomly selected label produced by the company will contain a defective compact disk is 4.28%.

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