Consider that a product be sold in batches of 123 pieces such as:

• Batch without inspection
• Inspected batch.

If the batch is inspected, it can be classified as:

• First quality
• Second-quality.

For inspection, 33 pieces are removed at random from the lot, without replacement.

• If more than 4 defective parts are found, the batch is considered second quality;
• Otherwise, the batch is classified as top quality.

Thus, the following values are made for the parts:

• If the batch is not inspected, each piece is sold for $968;
• If the batch is inspected and classified as second quality the part is sold for $790;
• If the batch is inspected and classified as first quality the part is sold for $1213.

Consider E[Y] = 34.44, where Y is expected value that represents the number of defective parts in a randomly chosen batch.

a) Assuming that the amount of defect calculated in the previous item occurs, rounded to a whole number, obtain the following probability value for P(X <= 4).

b) Obtain the probabilities associated with the batch being classified as first or second quality, according to the table.

Classification | First quality | Second quality

Probability | |

c) Calculate the probability distribution of the random variable C = "batch cost", as shown in the table.

Cost (C) | $1213 | $790

p(C) | |

d) What is the expected value of cost when the piece is sold on the inspected lot?

E[C]=