Three times each day, a quality engineer samples a component from a recently manufactured batch and tests it. Each part is classified as conforming (suitable for its intended use), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). An experiment consists of recording the categories of the three parts tested in a particular day.

Question

a. List the 27 outcomes in the sample space.
b. Let A be the event that all the parts fall into the same category. List the outcomes in A.
c. Let B be the event that there is one part in each category. List the outcomes in B.
d. Let C be the event that at least two parts are conforming. List the outcomes in C.
e. List the outcomes in .
f. List the outcomes in .
g. List the outcomes in .
h. List the outcomes in .
i. Are events A and C mutually exclusive? Explain.
j. Are events B and C mutually exclusive? Explain.

Answer 1

(a) The outcomes are the 27 different strings of 3 chosen from conforming (C), downgraded (D), and scrap (S). These are {CCC, CCD, CCS, CDC, CDD, CDS, CSC, CSD, CSS, DCC, DCD, DCS, DDC, DDD, DDS, DSC, DSD, DSS, SCC, SCD, SCS, SDC, SDD, SDS, SSC, SSD, SSS}.



(b) A = {CCC, DDD, SSS}



(c) B = {CDS, CSD, DCS, DSC, SCD, SDC}



(d) C = {CCD, CCS, CDC, CSC, DCC, SCC, CCC}



(e) The only outcome common to A and C is CCC. Therefore A? C = {CCC}.



(f) The set contains the outcomes that are either in A, in B, or in both. Therefore A ?B = {CCC, DDD, SSS, CDS, CSD, DCS, DSC, SCD, SDC}.



(g) contains the outcomes that are not in C. A? contains the outcomes that are in A but not in C. Therefore A? = {DDD, SSS}.



(h)

contains the outcomes that are not in A. ? C contains the outcomes that are in C but not in A. Therefore ? C = {CCD, CCS, CDC, CSC, DCC, SCC}.



(i) No. They both contain the outcome CCC.



(j) Yes. They have no outcomes in common.