Consumer’s Risk & Producer’s Risk

Population is unknown and population parameters are also unknown. In case of industrial applications, knowing the population is a very expensive and time consuming process. So a sampling scheme is designed. We base our judgment on the basis of a sample. This involves a risk of wrongfully rejecting a good lot and wrongfully accepting a bad lot. The prior is called producer’s risk and later is called consumer’s risk. With probability we can quantify these risks. Yesterday’s example illustrated the quantification of producer’s risk with the value of probability. These concepts are discussed further in this example.

Example:  A lot contains 52 integrated chips (IC). A sample of 4 IC is selected at random from each lot. According to the sampling scheme designed for quality control, if the sample contains more than 2 defectives it is rejected. Due to the choice of a bad supplier suppose the number of defective in a lot of 52 rise to 12 (this is unknown to us). What is the probability that this lot is still accepted?

Solution: Let X be the number of defective IC in a lot of 4 defectives.

P(lot is accepted) = P( no of defectives less than or equal to 2) = P(X=0) + P(X = 1) + P(X=2)

= [C(40, 4)/C(52, 4)]+[C(12, 1)C(40, 3)/C(50, 4)]+ [C(12, 2)C(40, 2)/C(50, 4)]

= 0.96

There is 96% chance that we can accept a lot with more than 10% of defective. Our objective is to have only 10% defectives in the population. This is consumer’s risk as consumers can get these defective products. The producer’s risk was quantified as 0.017% in yesterday’s example.