Question

It is common in many industrial areas to use a filling machine to fill boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and indeed they may A, fill to specification, B, underfill, and C, overfill. Generally the practice of underfilling is that which one hopes to avoid. Let P(C) = 0.052 while P(A) = 0.940. (a) What is the probability that the box is underfilled, P(B)? (b) Find P(A ∩ B). (c) Are A and B mutually exclusive events? Why or why not? (d) Find P(A ∪ B). (e) What is the probability that the machine does not overfill? (f) What is the probability that the machine either overfills or underfills?

Answer 1

(a) P(B) = 0.008, (b) P(A∩B) = 0, (c) Yes, A and B are mutually exclusive events, (d) P(A∪B)=0.948, (e) 0.948, (f) 0.06

Explanation:

We have three different posibilities

A: fill to specification

B: underfill

C: overfill

in probability the sum of the different events which are mutually exclusive should sum to 1, so, we should have

(a) P(B) = 1 - P(A)-P(C) = 1-0.940-0.052=0.008

(b) P(A∩B)=probability that the machine fill to specification and underfill = 0 because a machine can't fill to specification and underfill at the same time

(c) Yes, A and B are mutually exclusive events, because a machine can't fill to specification and underfill at the same time

(d) Because A and B are mutually exclusive events we should have that

P(A∪B)=P(A)+P(B)=0.940+0.008=0.948

(e) The probability that the machine does not overfill is the same that the probability that the machine fill to specification plus the probability that the machine underfill, i.e, the probability that the machine does not overfill is P(A)+P(B)=0.948, because does not overfill is equivalent either to fill to specification or to underfill.

(f) The probability that the machine either overfill or underfills is

P(C∪B)=P(C)+P(B)=0.052+0.008=0.06 because C and B are mutually exclusive events.