Step-by-step explanation:
Hello!
Three samples of components manufactured are taken per day. They are classified as:
D: "Conforming (suitable for its use)"
E: "Downgraded (unsuitable for the intended purpose but usable for another purpose)"
F: "Scrap (not usable)"
This classification includes the three events that may occur in your sample space S.
The experiment consists in recording the categories of the three parts tested in a day.
a. List the 27 outcomes in the sample space.
The possible outcomes in the space sample are the combinations of the three events. To avoid using the same letters as in the following questions I've named the evets as D, E, and F
S={DDD, DED, DFD, DEF, DFE, DEE, DFF, DDE, DDF , EDE, EEE, EFE, EED, EEF, EDF, EFD, EDD, EFF , FDF, FEF, FFF, FFE, FFD, FDE, FED, FDD, FEE}
b. Let A be the event that all the parts fall into the same category. List the outcomes in A.
A: "All the parts fall into the same category"
You have three possible outcomes for this event, that the three compounds are conforming, "DDD", that the three are unconforming, "EEE", or that the three compounds are scrap, "FFF". There are only three possible outcomes for this event.
S={DDD, EEE, FFF}
c. Let B be the event that there is one part in each category. List the outcomes in B.
B: "There is a part in each category"
This means, for example, The first one is conforming "D", the second one is unconforming "E" and the third one is scrap "F", then the first one may be unconforming "E", the second one is conforming "D" and the thirds one is scrap "F", and so on, you have 6 possible outcomes for this event:
S={DEF, DFE, EDF, EFD, FDE, FED}
d. Let C be the event that at least two parts are conforming. List the outcomes in C.
C: "At least two parts are conforming"
For this event, you can have two of the compounds to be considered conforming or the three of them.
S={DDD, DED, DFD, DDE, DDF , EDD, FDD}
A total of 7 combinations fit this event.
e. List the outcomes in A ∩ C
A ∩ C is an intersection between the event A and C, this means that there must be outcomes that are shared by both events.
Possible outcomes for A: S={DDD, EEE, FFF}
Possible outcomes for C: S={DDD, DED, DFD, DDE, DDF , EDD, FDD}
As you can see there is only one possible outcome shared by these two events. So the possible outcomes for A ∩ C are:
S= {DDD}
f. List the outcomes in A U B
A U B is the union between these two events, to see what outcomes this union has you have to add every outcome of A plus every outcome of B minus the possible outcomes that A and B share:
Possible outcomes for A: S={DDD, EEE, FFF}
Possible outcomes of B: S={DEF, DFE, EDF, EFD, FDE, FED}
The possible outcomes for A U B are:
S={DDD, EEE, FFF, DEF, DFE, EDF, EFD, FDE, FED, EEF, EDF, EFD, }
g. List the outcomes in A ∩ c
c is the complementary event of C, it could also be symbolized as 
If C: "At least two parts are conforming" then its complemental event will be
: At most one part is conforming"
This means that one or none parts are conforming, and it's possible outcomes are:
S= {DEF, DFE, DEE, DFF, EDE, EEE, EFE, EED, EFF , FDF, FEF, FFF, FFE, FFD, FDE, FED, FEE}
Possible outcomes for A: S={DDD, EEE, FFF}
As you see there are two events on "A" that also appear in the definition of ""
The possible outcomes for A ∩ are:
S= {EEE, FFF}
h. List the outcomes in Ac ∩ C
Ac is the complementary event of A, also symbolized as 
If A: "All the parts fall into the same category", then its complemental event will be
: "Not all the parts fall into the same category"
and its possible outcomes are the remaining 24 occurrences:
S={DED, DFD, DEF, DFE, DEE, DFF, DDE, DDF , EDE, EFE, EED, EEF, EDF, EFD, EDD, EFF , FDF, FEF, FFE, FFD, FDE, FED, FDD, FEE}
Possible outcomes for C: S={DDD, DED, DFD, DDE, DDF , EDD, FDD}
As you can see the events and C share 6 occurrences in common, so the possible outcomes for the intersection will be:
S= {DED, DFD, DDE, DDF , EDD, FDD}
i. Are events A and C mutually exclusive? Explain.
Two events are mutually exclusive when the occurrence of one of them keeps the other from occurring, i.e. it can happen one or the other but not both.
A and C are not mutually exclusive since the three pieces may have the same category, "D: conforming" and at least two of them to be conforming "DD-" at the same time in the occurrence "DDD" and mutually exclusive events never happen at the same time.
ii. Are events B and C mutually exclusive? Explain.
B and C are mutually exclusive, you can easily see this if you compare the possible outcomes of both events:
Possible outcomes of B: S={DEF, DFE, EDF, EFD, FDE, FED}
Possible outcomes for C: S={DDD, DED, DFD, DDE, DDF , EDD, FDD}
There are no shared elements by these events. This means that if you were to take three pieces randomly sampled in one day and fit the definition of B, then they will not fir the definition of C and vice versa.
I hope it helps!
