4. Probability Rules and Conditional
Probability
4.1
General Methods
In the mathematical de
fi
nition of probability, an arbitrary event
A
is merely some subset of the sample
space
S
. The following rules hold:
1.
P
(
S
) = 1
2.
For any event
A,
0
≤
P
(
A
)
≤
1
It is also obvious from our de
fi
nitions in Chapter 2 that if
A
and
B
are two events with
A
⊆
B
(that
is, all of the simple events in
A
are also in
B
), then
P
(
A
)
≤
P
(
B
)
.
It is often helpful to use elementary ideas of set theory in dealing with probability; as we show
in this chapter, this allows certain rules or propositions about probability to be proved. Before going
on to speci
fi
c rules, we’ll review Venn diagrams for sets. In the drawings below, think of all points
in
S
being contained in the rectangle, and those points where particular events occur being contained
in circles. We begin by considering the union
(
A
∪
B
)
, intersection
(
A
∩
B
)
and complement
(
¯
A
)
of
sets (see Figure 4.3). At the URL http://stat-www.berkeley.edu/users/stark/Java/Venn.htm, there is an
interesting applet which allows you to vary the area of the intersection and construct Venn diagrams for
a variety of purposes.
Example:
Suppose for students
fi
nishing 2A Math that 22% have a math average
≥
80%, 24% have a STAT 230
mark
≥
80%, 20% have an overall average
≥
80%, 14% have both a math average and STAT 230
≥
80%, 13% have both an overall average and STAT 230
≥
80%, 10% have all 3 of these averages
≥
80%, and 67% have none of these 3 averages
≥
80%. Find the probability a randomly chosen math
student
fi
nishing 2A has math and overall averages both
≥
80% and STAT 230
<
80%.
Solution:
When using rules of probability it is generally helpful to begin by labeling the events of
interest.
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