Question

Acceptance Sampling. With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is accepted if every item in the sample is found to be okay. Exercises 27 and 28 involve acceptance sampling.

Something Fishy: The National Oceanic and Atmospheric Administration (NOAA) inspects seafood that is to be consumed. The inspection process involves selecting seafood samples from a larger “lot.” Assume a lot contains 2875 seafood containers and 288 of these containers include seafood that does not meet inspection requirements. What is the probability that 3 selected container samples all meet requirements and the entire lot is accepted based on this sample? Does this probability seem adequate?

Short answer

The probability that the sample will be accepted is equal to 0.728.

Yes, the above probability seems adequate as it is sufficiently high for the acceptance of the entire lot.

Step-by-step solution

Given information

The number of seafood containers in a lot is 2875. Out of these, 288 do not meet the requirements.

The three samples are selected without replacement.

Multiplication rule of probability

The probability that n number of different events, independent in nature, occur simultaneously is represented and computed as follows:

$P\left(A_{1}and{A}_{2}and{A}_{3}and...and{A}_n\right)=P\left(A_1\right)\times P\left(A_2\right)\times P\left(A_3\right)\times ...\times P\left(A_n\right)$

Here, $P\left(A_2\right),P\left(A_3\right)$, etc., are successive probabilities that depend upon the occurrences of the previous events.

Compute the counts of events in the study

The entire batch is accepted if all three containers meet the requirements.

Let A be the event of selecting the first container which meets the requirements.

Let B be the event of selecting the second container which meets the requirements.

Let C be the event of selecting the third container which meets the requirements.

The total number of containers in the lot is equal to 2875.

The number of containers that do not meet the inspection requirements is equal to 288.

The number of containers that meet the inspection requirements is:

$2875-288=2587$

As the selection is done without replacement, the probability of successive containers which meets the requirement changes as follows:

$$\begin{gathered} P\left(A\right)=\frac{2587}{2875} \\ P\left(B\right)=\frac{2586}{2874} \\ P\left(C\right)=\frac{2585}{2873} \end{gathered}$$

The probability of selecting three containers of acceptable quality from the lot is given as follows:

$$\begin{gathered} P\left(A\right)=\frac{2587}{2875}\times \frac{2586}{2874}\times \frac{2585}{2873} \\ =0.728 \end{gathered}$$

Therefore, the probability of selecting three containers that meet the requirements is equal to 0.728.

Interpret the adequacy of the result

The probability of accepting the lot, when the sample of three containers is selected, is 0.728.

As the probability value is high, it seems to be adequate for the acceptance of the entire lot of 2875 containers.