CS 2800: Discrete Math
Oct. 19, 2011
Lecture
Lecturer: John Hopcroft
Scribe:Doo San & June
Review
Note
: the next homework will have a thought problem at the bottom of each homework. No credit given
for answer.
Basic Probability Terms
These are the terms you need to know to understand the basic principles of probability.
Probability Space : All possible outcomes
Probability Mass Function : The probability that the outcome
i
happens
Event : A subset of Probability Space
Example
Let’s think about the situation that we are rolling a die.
Probability Space :
S
=
{
1
,
2
,
3
,
4
,
5
,
6
}
Probability Mass Function :
f
(
i
) =
1
6
Event :
{
1
,
2
,
3
}
Note
1. Probability of a particular event
E
:
P
(
E
) =
∑
i
∈
E
f
(
i
)
2. Birthday problem can be understood with the similar concept with collisions in hashing.
It can be
roughly worked out to
√
n
, where
n
is the number of your buckets. In the birthday problem, it would
be 365 buckets, or roughly 20 people until someone has the same birthday.
Continuous Probability
In the situation described above, there are a finite number of outcomes with weight,
f
(
i
)
>
0. We call these
probabilities discrete.
Now, we need to think about infinite sample spaces  i.e.
spaces where an infinite
number of events could happen. In this case, each point source must have weight 0. Before we move on, let’s
think about why. We can assume that the outcomes in this case are all equally likely. The probability of
each of these events can be formulated as
1
N
(
S
)
=
1
∞
= 0. Therefore, we need to introduce the new concept
of probability density function of which the area under the curve is the probability of the outcomes in the
certain range.
The difference in probability mass function
f
(
x
) and the probability density function
P
(
x
) can be sum
marized as below.
1.
f
(
i
) : The probability that the particular outcome
i
happens
2.
b
R
a
P
(
x
)
dx
: The probability that the outcomes between
a
and
b
happen.
1
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3.
F
(
x
) =
x
R
∞
P
(
x
)
dx
: Cumulative distribution function; the probability that all the outcomes less than
x
happen. In this sense,
b
R
a
P
(
x
)
dx
can be reformulated as
F
(
b
)

F
(
a
).
With all the prior knowledge mentioned above, now we can move onto the event in a certain range. First
of all, we can march through the example of the rationals in the range 0 to 1.
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 '07
 SELMAN
 Probability, Probability theory, Lecturer, John Hopcroft

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