2.2. CONTINUOUS DENSITY FUNCTIONS
57
game there may well be!), it is natural to assume that the coordinates are chosen
at random.
(When doing this with a computer, each coordinate is chosen uniformly
from the interval [

1
,
1]. If the resulting point does not lie inside the unit circle,
the point is not counted.) Then the arguments used in the preceding example show
that the probability of any elementary event, consisting of a single outcome, must
be zero, and suggest that the probability of the event that the dart lands in any
subset
E
of the target should be determined by what fraction of the target area lies
in
E
. Thus,
P
(
E
) =
area of
E
area of target
=
area of
E
π
.
This can be written in the form
P
(
E
) =
±
E
f
(
x
)
dx ,
where
f
(
x
) is the constant function with value 1
/π
. In particular, if
E
=
{
(
x, y
) :
x
2
+
y
2
≤
a
2
}
is the event that the dart lands within distance
a <
1 of the center
of the target, then
P
(
E
) =
πa
2
π
=
a
2
.
For example, the probability that the dart lies within a distance 1/2 of the center
is 1/4.
±
Example 2.9
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 Spring '09
 scf
 Probability theory, probability density function, dart lands

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