Download probability theory

PROF. VINOD CLASS NOTES FOR PROBABILITY THEORY
Randomness in tossing a coin is easily described.
Not so with respect to randomness of a stock market.
Prob. Theory is designed to understand and tame randomness.
Sample space= set of all distinct outcomes
S={1, 2, 3, 4, 5, 6} is sample space for tossing a die.
Simple event= any member of S. e.g. {4} is a simple event.
(Regular) Event=a set of simple events (set of points of S)
e.g. set of even numbers {2,4,6} can be an event.
Prob. of an Event =
number  of  favorable  po int s  describing  the  event
total  number  of  po int s  in  the  sample  space
Note that in the formula above, the Denominator equals the total no. of mutually exclusive,
exhaustive and equally likely points.
P(E) = #(E) / #(S) or n(E) / n(S) is called classical probability rule in software lesson 4.1
Lord Keynes (economist) gave a relative frequency interpretation of probability in his Treatise on
Probability published in 1920's.
LAWS of PROB.:
1) P(A)=0 means event A cannot happen.
2) P(A)=1 means event A MUST happen.
3) 0  P(A)  1
P(A) cannot be negative or larger than one.
4) sum of probabilities of simple events is one.
e.g. P(Head)+P(Tail)=1
Compound event is defined by combining two or more events.
Union of events A and B is the set of outcomes included
in both A and B and is denoted by A U B
Intersection is the set of outcomes in both A and B (at the same time)
Complement of an event A is a set of outcomes in Sample
space S which is not in A
Two sets are mutually exclusive if they have no points in common
Prob. LAW 5) Prob. of complement is P( Ac)= 1 - P(A)
if A denotes the event and Ac denotes the complement
e.g. A={2,4,6} then Ac={1,3,5}
Prob. LAW 6) Addition Rule if A and B are mutually exclusive:
P( A or B )=P(A U B)=P(A)+P(B)
Example of misuse of addition rule.
Insurance Underwriters have established that the probability of city experiencing disasters in the
next five years is
0.3 for a Tornado, 0.4 for Hurricane,
0.2 for an Earthquake, and 0.4 for Flooding.
True or False?
P( T or H or E or F )=0.3+0.4+0.2+0.4=1.3
False.
Prob. < or = 1 fails.
Do not use addition rule when the events are not mutually exclusive. Two or more disasters can
occur over the five year period, and some may occur more than once. So Law 6 cannot be used!
The general addition rule is known as Law 7.
LAW 7) don’t forget the last term with minus sign
P(A union B)=P(A U B)=P(A)+P(B)-P(A  B)
Conditional Prob.= Prob. that an event will occur given that
some other event has either already occurred or is certain to occur.
LAW 8) Conditional Prob. of A given that B has occurred
is P ( A | B ) = P ( A  B) / P (B)
Statistical Independence: Two events A and B are independent
if and only if P(A|B)=P(A)
or P(B |A)=P(B)
which says that conditional prob. equals unconditional prob.,
which is as if the condition did not matter
Example: A= suite of clubs, Sample space is 52 cards
event B= Aces or Kings
P(B | A) = prob of an Ace or a King given that the card is from the suite of
clubs.
P( A  B) = (joint prob of A and B at the same time)
= (common points in A and B are Ace of clubs and King of
clubs, or two points so the joint prob is 2/52
P(A)=( no of favorable points for event A)/(points in S)=13/52= marginal or
unconditional probability
Conditional prob. of B= P(B|A)= P( A interset B) / P(A)
= (joint prob)/(unconditional prob)= (2/52) / (13/52) =(2/13)
Unconditional prob of B = P(B)= 8/52 ( note 4 aces and 4 kings)
Since conditional prob. of B (=2/13) equals the uncondition prob.(8/52)
THis means that A and B are statistically INDEPENDENT.
LAW 9)
Multiplication Rule if A and B are independent:
P( A and B together)=P(A  B)=P(A)P(B)
Example of misuse of multiplication rule:
A lady was mugged by a couple. One interracial couple
(p=.001), blond girl (p=0.25) with ponytail (p=0.1),
bearded (p=0.1) black man (p=1/3), who drove a yellow
car (p=0.1), was convicted by circumstantial evidence.
A random couple with all six characteristic has the
prob.=0.001 x 0.25 x 0.1 x 0.1 x (1/3) x 0.1
= 1/(12,000,000). Using this argument a couple was
convicted.
Higher court overturned the conviction because the
multiplication rule is used without checking whether
independence assumption holds true. In general the
multiplication rule is
P(A  B)=P(A)P(B|A)
Fundamental Counting Principle
Consider a sequence of n experiments where the first experiment has k1 outcomes, second has k2
outcomes and so on.
Total number of possible outcomes for the sequence of experiments
is given by the simple multiplication: k1*k2….*kn
Factorials
A factorial of n (n!) is a product all positive integers less than or equal to n.
(just multiply all integers 1,2,3 etc till n)
Permutations (when order is important) nPr = n! / (nr)!
n is number of individuals E.g. 8
r=2 positions to be filled President and Secretary, Ans 8P2=8!/6! = 8*7 =56
factorial(8)/factorial(8-2) is the R command. Order Matters
Combinations (when order is NOT AT ALL important) nCr = n! / ( r! * [n-r]!)
R command is choose(n,r)
Mississippi example in lesson
Permutations when k1 are all alike, k2 are all alike and so on till kp are all alike and k1 to kp add
up to n
n! / ( k1! * k2! * . . . *kp! )