Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
1
Counting Methods
We often find the probability of an event by counting the number of elements
in a simple sample space.
Basic methods for counting are:
Permutations
Combinations
Permutation
An arrangement of r distinct objects in a definite order is c
Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
Introduction to the Probability Theory
The Basic Terminology
E
Experiment
The experiment describes the problem to solve.
s
Outcome
The outcome varies each time we repeat the experiment.
S
Sample Space
The set of all possible outcomes. s S always.
A
Event
Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
1
Stirlings Formula
a
n ! ~ 2 n
n
1
2 e n
indiccates the ratioo of the two sides
s
tends too unity n .
Primerr
Maclaurrin Series
log 1 x x
x 2 x3 x 4
2
3
4
1 x 1
p1
repllacing x by x
x 2 x3 x 4
1
x
2
3
1 x
4
addding eq.p1 annd eq.p 2,
log
p2
1
1 x
x3
Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
1
Counting Methods Variations
Some problems may seem different. But they turn out to have an identical solution,
Binomial
r
x y
r
r
j x j yr j
j 0
For the special case of x y 1,
2r
r
r
j
j 0
For the special case of x p and y 1 p, for any r and p,
1
Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
1
Coin Tossing Games
1) Game of tossing one coin.
Keep tossing a coin until A or B wins. A wins on contiguous HH, and B wins on HT.
Ans. P[ A wins ] P[ B wins ] 1 / 2 .
2) Game of tossing one coin.
Keep tossing a coin until A or B wins. A wins on contiguo
Korea Advanced Institute of Science and Technology
probability and random process
EE 528

Winter 2014
1
Union of Events
Consider events A1, A2 , , AN .
The union of the events is A A1 A2 AN .
We want to find P A .
For N 2,
P A P A1 P A2 P A1 A2
For N 3,
P A P A1 P A2 P A3
P A1 A2 P A1 A3 P A2 A3
P A1 A2 A3
For generalization to an arbitrary number N