Cryptography Reference
In-Depth Information
Pr[Ω] is conventionally set to one, and Pr[
∅
] is set to zero. Furthermore, one
frequently needs the complement of an event
A
. It consists of all elements of Ω that
are not elements of
A
. The complement of
A
is denoted as
A
, and its probability
can be computed as follows:
]=
ω∈
Ω
\A
Pr[
A
Pr[
ω
]
If we know Pr[
A
], then we can easily compute
Pr[
A
]=1
−
Pr[
A
]
because Pr[
A
] and Pr[
A
] must sum up to one.
$;
<
$;
<
Figure 4.1
A discrete probability space.
A discrete probability space is illustrated in Figure 4.1. There is a sample
space Ω and a probability measure Pr[
·
] that assign a value between 0 and 1 to every
elementary event
ω
Ω.
If, for example, we want to compute the probability of the event that, when
flipping five coins, we get three heads, then the sample space is Ω=
∈
Ω or event
A⊆
5
and
the probability distribution is uniform. This basically means that every element
{
1
,
0
}
