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Bender Electronics buys keyboards for its computers from another company. The keyboards are received in shipments of 100 boxes, each box containing 20 keyboards. The quality control department at Bender Electronics first randomly selects one box from each shipment and then randomly selects 5 keyboards from that box. The shipment is accepted if not more than 1 of the 5 keyboards is defective. The quality control inspector at Bender Electronics selected a box from a recently received shipment of keyboards. Unknown to the inspector, this box contains 6 defective keyboards. a. What is the probability that this shipment will be accepted? b. What is the probability that this shipment will not be accepted?

Step by step solution

01

Determine the Known Variables

It is given that a box contains 20 keyboards and out of these, 6 are defective. This means the probability of choosing a defective keyboard (\(p\)) is 6/20 or 0.3. The problem states that 5 keyboards are tested, so the number of trials (\(n\)) is 5. The shipment is accepted if not more than 1 of the 5 keyboards is defective. Therefore, an acceptable outcome (success) is defined when 0 or 1 keyboard is defective.

02

Calculate the Probability for no Defective Keyboards

To find the probability that none of the keyboard is defective we use the binomial probability formula: \(P(0) = C(5, 0) * (0.3)^0 * (1 - 0.3)^(5-0)\). The calculation gives us the probability that none of the selected keyboards is defective.

03

Calculate the Probability for one Defective Keyboard

To find the probability that one of the keyboard is defective we use the binomial probability formula: \(P(1) = C(5, 1) * (0.3)^1 * (1 - 0.3)^(5-1)\). The calculation gives us the probability that one of the selected keyboards is defective.

04

Calculate the Probability for Shipment Acceptance

The shipment will be accepted if either no keyboard or 1 keyboard is defective. We can get this probability by adding the individual probabilities calculated in steps 2 and 3. \(P(Accepted) = P(0) + P(1)\). This provides the probability that the shipment will be accepted.

05

Calculate the Probability for Shipment Rejection

Since the shipment will be rejected if more than 1 keyboard is defective, which is the opposite event of the shipment being accepted. Therefore, we can subtract the probability of shipment acceptance from 1 to get the probability of shipment rejection. \(P(Rejected) = 1 - P(Accepted)\). This gives the probability that the shipment will not be accepted.

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