Cars are being produced by two factories, but factory I produces twice as many cars as...

Cars are being produced by two factories, but factory I produces twice as many cars as factory II in a given time. Factory I is known to produce 2% defectives and factory II produces 1% defectives. A car is examined and found to be not defective. What is the probability that it came from factory 1?

Solutions

Expert Solution

P[ car produced in factory I ] = 2/3

P[ car produced in factory II ] = 1/3

P[ defective | factory I ] = 2% = 0.02

P[ non defective | factor I ] = 1 - P[ defective | factory I ]

P[ non defective | factor I ] = 1 - 0.02

P[ non defective | factor I ] = 0.98

P[ defective | factory II ] = 1% = 0.01

P[ non defective | factor II ] = 1 - P[ defective | factory II ]

P[ non defective | factor II ] = 1 - 0.01

P[ non defective | factor II ] = 0.99

P[ non defective ] = P[ non defective | factor II ]*P[ car produced in factory II ] + P[ non defective | factor I ]*P[ car produced in factory ]

P[ non defective ] = 0.99*(1/3) + 0.98*(2/3)

P[ non defective ] = ( 0.99 + 1.96 )/3

P[ non defective ] = 0.983

We need to find

P[ factor I | non defective ] = P[ non defective | factor I ]*P[ car produced in factory ] / P[ non defective ]

P[ factor I | non defective ] = 0.98*(2/3) / 0.983

P[ factor I | non defective ] = 0.653/0.983

P[ factor I | non defective ] = 0.6644

orchestra answered 1 year ago

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