1.2.
DISCRETE PROBABILITY DISTRIBUTIONS
25
First toss
Second toss
Third toss
Outcome
H
H
H
H
H
H
T
T
T
T
T
T
(Start)
ω
ω
ω
ω
ω
ω
ω
ω
1
2
3
4
5
6
7
8
H
T
Figure 1.8: Tree diagram for three tosses of a coin.
Let
A
be the event “the first outcome is a head,” and
B
the event “the second
outcome is a tail.” By looking at the paths in Figure 1.8, we see that
P
(
A
) =
P
(
B
) =
1
2
.
Moreover,
A
∩
B
=
{
ω
3
,
ω
4
}
, and so
P
(
A
∩
B
) = 1
/
4
.
Using Theorem 1.4, we obtain
P
(
A
∪
B
)
=
P
(
A
) +
P
(
B
)

P
(
A
∩
B
)
=
1
2
+
1
2

1
4
=
3
4
.
Since
A
∪
B
is the 6element set,
A
∪
B
=
{
HHH,HHT,HTH,HTT,TTH,TTT
}
,
we see that we obtain the same result by direct enumeration.
In our coin tossing examples and in the die rolling example, we have assigned
an equal probability to each possible outcome of the experiment. Corresponding to
this method of assigning probabilities, we have the following definitions.
Uniform Distribution
Definition 1.3
The
uniform distribution
on a sample space
Ω
containing
n
ele
ments is the function
m
defined by
m
(
ω
) =
1
n
,
for every
ω
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 Spring '09
 scf
 Probability distribution, Probability theory, probability density function, Coin flipping, Coin

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