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In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________
Correct answer is between '0.88,0.95'. Can you explain this answer?
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In a production line of a factory, each packet contains four items. Pa...
Problem: In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________
Solution:
Given,
- The probability of an item being defective is 20%.
- The quality manager approves the packet for shipment if at most one item in the packet is found to be defective.
- We need to find the probability that out of the three randomly inspected packets at least two are approved for shipment.
Step 1: Probability of one item being defective
The probability of one item being defective is given as 20%. Hence, the probability of one item being non-defective is (100-20)% = 80%.
Step 2: Probability of a packet being approved
The quality manager approves the packet for shipment if at most one item in the packet is found to be defective. Hence, the probability of a packet being approved is given as follows:
- Probability of all four items being non-defective: (0.8)^4 = 0.4096
- Probability of exactly one item being defective: 4C1 x (0.2) x (0.8)^3 = 0.4096
- Probability of a packet being approved = 0.4096 + 0.4096 = 0.8192
Step 3: Probability of at least two packets being approved
We need to find the probability that out of the three randomly inspected packets at least two are approved for shipment. There can be three cases:
- Case 1: All three packets are approved
- Case 2: Exactly two packets are approved
- Case 3: None of the packets are approved
Probability of Case 1:
- Probability of one packet being approved = 0.8192
- Probability of all three packets being approved = (0.8192)^3 = 0.551
Probability of Case 3:
- Probability of one packet being rejected = 1 - 0.8192 = 0.1808
- Probability of all three packets being rejected = (0.1808)^3 = 0.006
Probability of Case 2:
- Probability of two packets being approved and one packet being rejected = 3C2 x (0.8192)^2 x (0.1808) = 0.364
Hence, the probability of at least two packets being approved = Probability of Case 1 + Probability of Case 2 = 0.551 + 0.364 = 0.915 (rounded off to two decimal places)
Therefore, the probability that out of the three randomly inspected packets at least two are approved for shipment equals 0.915.
Solution:
Given,
- The probability of an item being defective is 20%.
- The quality manager approves the packet for shipment if at most one item in the packet is found to be defective.
- We need to find the probability that out of the three randomly inspected packets at least two are approved for shipment.
Step 1: Probability of one item being defective
The probability of one item being defective is given as 20%. Hence, the probability of one item being non-defective is (100-20)% = 80%.
Step 2: Probability of a packet being approved
The quality manager approves the packet for shipment if at most one item in the packet is found to be defective. Hence, the probability of a packet being approved is given as follows:
- Probability of all four items being non-defective: (0.8)^4 = 0.4096
- Probability of exactly one item being defective: 4C1 x (0.2) x (0.8)^3 = 0.4096
- Probability of a packet being approved = 0.4096 + 0.4096 = 0.8192
Step 3: Probability of at least two packets being approved
We need to find the probability that out of the three randomly inspected packets at least two are approved for shipment. There can be three cases:
- Case 1: All three packets are approved
- Case 2: Exactly two packets are approved
- Case 3: None of the packets are approved
Probability of Case 1:
- Probability of one packet being approved = 0.8192
- Probability of all three packets being approved = (0.8192)^3 = 0.551
Probability of Case 3:
- Probability of one packet being rejected = 1 - 0.8192 = 0.1808
- Probability of all three packets being rejected = (0.1808)^3 = 0.006
Probability of Case 2:
- Probability of two packets being approved and one packet being rejected = 3C2 x (0.8192)^2 x (0.1808) = 0.364
Hence, the probability of at least two packets being approved = Probability of Case 1 + Probability of Case 2 = 0.551 + 0.364 = 0.915 (rounded off to two decimal places)
Therefore, the probability that out of the three randomly inspected packets at least two are approved for shipment equals 0.915.
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In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer?
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In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer?.
In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In a production line of a factory, each packet contains four items. Past record shows that 20% of the produced items are defective. A quality manager inspects each item in a packet and approves the packet for shipment if at most one item in the packet is found to be defective. Then the probability (round off to 2 decimal places) that out of the three randomly inspected packets at least two are approved for shipment equals __________Correct answer is between '0.88,0.95'. Can you explain this answer?.
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