Answer to Question #264952 in Statistics and Probability for Kim

Question

. (79 SEPT ME BOARD) A certain manufactured part can be defective because it has one or more out of the three possible defects: insufficient tensile strength, a burr, or a diameter outside of tolerance limit. In a lot of 500 pieces; 19 have a tensile strength defects, 17 have a burr, 11 have an unacceptable diameter, 12 have tensile strength and burr defects, 7 have tensile strength and diameter defects, 5 have burr and diameter defects and 2 have all three defects. How many of the pieces have no defects

Answer 1

Let A be the set of parts that has only insufficient tensile strength, B be the set of parts that has only a burr, C be the set of the parts that has only diameter outside of tolerance limit. Then

$|A∪B∪C|=|A|+|B|+|C|-|A⋂B|-|A⋂C|-|C⋂B|+|A⋂B⋂C|=19+17+11-12-7-5+2=25$

We know that 25 details out of 500 has defects, so there is 500 - 25 = 475 pieces without any defects