Download A1 ∪ A2 ∪ A3 = the event that at least one plant is completed by the

ENGR 323
D. Stier
Beautiful Homework #2
Problem 2-8
page 1 of 3
Problem 2-8
An engineering construction firm is currently working on power plants at three different sites. Let Ai
denote the event that the plant at site i is completed by the contract date. Use the operations of union,
intersection, and complementation to describe each of the following events in terms of A1, A2, A3, draw a
Venn diagram, and shade the region corresponding to each one.
Some Helpful Definitions For Set Theory:
1.
The union of two events A and B, denoted by A ∪ B and read “A or B,” is the event consisting of all
outcomes that are either in A or in B or in both events (so that the union include outcome for which
both A and B occur as well as outcomes for which exactly one occurs) (Devore 46).
2.
The intersection of two events A and B, denoted by A ∩ B and read “A and B,” is the event consisting
of all outcomes that are in both A and B (Devore 46).
3.
The complement of an event A, denoted by A’ or Ac, is the set of all outcomes in the sample space (S)
that are not contained in A (Devore 46).
4.
When events A and B have no outcomes in common, they are said to be mutually exclusive or
disjoint events (Devore 47).
*note: the sizes of the circles used in the Venn diagrams are irrelevant in this problem.
Part a) At least one plant is completed by the contract date.
Solution: This statement implies that at least one plant is completed by the contract date at one of the three
sites. We can reword this to say the plant is completed at site 1 or site 2 or site 3 by the contract date. The
shading in Figure 1 below show this more clearly. Here is a symbolic representation of the answer:
A1=the event that the plant at site 1 is completed by the contract date
A2=the event that the plant at site 2 is completed by the contract date
A3=the event that the plant at site 3 is completed by the contract date
S =sample space (the set of all possible outcomes )
A1 ∪ A2 ∪ A3 =
the event that at least one plant is completed by the contract date
S
A2
A1
A3
Figure 1. A Venn diagram representing A1 ∪A2∪ A3
ENGR 323
D. Stier
Beautiful Homework #2
Problem 2-8
page 2 of 3
Part b) All plants are completed by the contract date.
Solution: The problem states that all plants are completed by the contract date. This implies that we are
looking for the set of outcomes that is contained in all three events , A1 and A2 and A3. In other words, we
can say the plant is completed at site 1 and site 2 and site 3, indicating for us to use intersections to
represent the event. The shaded area in the Figure 2 below covers the area where the three events intersect
or in other words, they all occur.
A1 ∩ A2 ∩ A3 =
the event all plants are completed by the contract date
S
A2
A1
A3
Figure 2. A Venn diagram representing A1 ∩ A2 ∩ A3
Part c) Only the plant at site one is completed by the contract date
Solution: In this statement event A1 occurs, but not A1 and some other event A2 or A3. We can think of
this event as all possible outcomes except those which include events A2 and A3. In probability terms, all
of the outcomes in the sample space including event A1, except events A2 and A3. The shaded area in
Figure 3 represents this event. The event can be represented in symbolic terms using the intersection of the
event A1 with the compliments of events A2 and A3:
A1 ∩ A2 ′ ∩A3′ =
the event A1 not including events A2 or A3
A2
A1
A3
Figure 3. A Venn diagram representing A1∩ A2 ′∩A3′
ENGR 323
D. Stier
Beautiful Homework #2
Problem 2-8
page 3 of 3
Part d) Exactly one plant is completed by the contract date
Solution: This statement indicates that only one of the events occurs. None of the outcomes that contain
more than one event can be included. This problem is similar to part c, but the condition is extended to
events A2 and A3. This event is shown in Figure 4 as the outcomes in the three events which don’t intersect
with any outcomes from different events. We can represent this event using unions, intersections, and
compliments:
(A1 ∩A2′∩A3′) ∪ ( A1′∩ A2 ∩ A3′) ∪ (A1′∩ A2′∩ A3) =
the event A1 not including events A2 and A3 or the event A2 not including
events A1 or A3 or the event A3 not including the events A1 or A2
S
A2
A1
A3
Figure 4. Venn diagram representing
(A1∩A2′∩A3′)∪( A1′∩ A2 ∩ A3′)∪ (A1′∩ A2′∩ A3)
Part e) Either the plant at site 1 or both of the other two plants are completed by the contract date.
Solution: We can rewrite this problem to say all possible outcomes that are either in event A1 or in events
A2 and A3. As Figure 5 shows, this area includes the outcomes in event A1 as well as the outcomes where
events A2 and A3 intersect. We can define this event as A1 in union with the intersection of A2 and A3:
A1 ∪ (A2 ∩ A3)=
the event that the plants at site A1 or A2 and A3 are completed by the contract
date
S
A2
A1
A3
Figure 5. Venn diagram representing A1∪(A2∩A3)