Counting plays a
crucial role in probability theory. One basic principle of counting is the Multiplication
Principle that allows counting objects that are constructed in successive
steps by multiplying together the numbers of ways of doing each step. As an
example: if there are 4 choices for an appetizer and four choices for a main
dish, the total number of dinners is 4 * 4 = 16. In order to count the total
number of objects in disjoint
sets of objects, Addition
Principle is used. As an example, suppose that within a set of strings, two
start with a and four start with b. Then 2 + 4 + 6 start with either a or b.
A permutation of n distinct elements x1, …, xn is an
ordering of n elements x1,…, xn. Using the Multiplication Principle, a
permutation of n (n!) elements can be constructed in n successive steps. n! =
N(n-1)(n-2)…2 * 1. A combination is a selection of objects without
regard to order. The number of r-combinations of n-element set is denoted C(n, r)
and is calculated by the formula: n!/(n
–r)!r!.
Discrete probability is used to compute the chance that an event
happens in a sample space. The probability P(E) is: P(E) = |E|/|S|.
Imagine A and B are events. Then:
P(A or B) = P(A) + P(B) – P(
A and B).
A probability given that some
event occurred is called a conditional probability. Let C and D be
events. The conditional probability of E given F is: P(C|D) = P(C and D)/P(D).
Pattern recognition
places items into various classes based on features of the items.
Bayes’ Theorem is useful in
computing the probability of a class given a set of features.
The binomial theorem can be
used to expand expressions. If a and b are real numbers and
n
n is a positive integer, then
(a+b)^n = ΣC(n, k)a^n-kb^k. The numbers C(n, k) are known
K = 0
binomial coefficients.
APPLICATIONS
1. Probability is used in the
field of Computational Statistics
http://www.clarkson.edu/~dobrowb/probweb/probweb.html
2. Probability is also used
in the design of Stochastic Networks
http://www.statslab.cam.ac.uk/~richard/stoch_nets/
QUESTIONS
2. Can anybody state a fourth
form of the Pigeonhole Principle?