Let S be a sample space and E and F be events associated with S . Suppose that Pr ( E ) = .5 , Pr ( F ) = .4 , and Pr ( E ∩ F ) = .1 . Calculate a. Pr ( E | F ) b. Pr ( F | E ) c. Pr ( E | F ′ ) d. Pr ( E ′ | F ′ ) .
Let S be a sample space and E and F be events associated with S . Suppose that Pr ( E ) = .5 , Pr ( F ) = .4 , and Pr ( E ∩ F ) = .1 . Calculate a. Pr ( E | F ) b. Pr ( F | E ) c. Pr ( E | F ′ ) d. Pr ( E ′ | F ′ ) .
Solution Summary: The author calculates the value of Pr(E|F) if the sample space is S and the events are E and F.
Let S be a sample space and E and F be events associated with S. Suppose that
Pr
(
E
)
=
.5
,
Pr
(
F
)
=
.4
,
and
Pr
(
E
∩
F
)
=
.1
. Calculate
a.
Pr
(
E
|
F
)
b.
Pr
(
F
|
E
)
c.
Pr
(
E
|
F
′
)
d.
Pr
(
E
′
|
F
′
)
.
Definition Definition For any random event or experiment, the set that is formed with all the possible outcomes is called a sample space. When any random event takes place that has multiple outcomes, the possible outcomes are grouped together in a set. The sample space can be anything, from a set of vectors to real numbers.
Give an example of a graph with at least 3 vertices that has exactly 2 automorphisms(one of which is necessarily the identity automorphism). Prove that your example iscorrect.
3. [10 marks]
Let Go (Vo, Eo) and G₁
=
(V1, E1) be two graphs that
⚫ have at least 2 vertices each,
⚫are disjoint (i.e., Von V₁ = 0),
⚫ and are both Eulerian.
Consider connecting Go and G₁ by adding a set of new edges F, where each new edge
has one end in Vo and the other end in V₁.
(a) Is it possible to add a set of edges F of the form (x, y) with x € Vo and y = V₁ so
that the resulting graph (VUV₁, Eo UE₁ UF) is Eulerian?
(b) If so, what is the size of the smallest possible F?
Prove that your answers are correct.
Let T be a tree. Prove that if T has a vertex of degree k, then T has at least k leaves.
Chapter 6 Solutions
Pearson eText for Finite Mathematics & Its Applications -- Instant Access (Pearson+)
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