4203 Mathematical ProbabilityChapter 2: Probability

Instructor: Dr. Ayona ChatterjeeSpring 2011

Classical Probability concept

If there are N equally likely possibilities of which one must occur and n are regarded as favorable, or as a success, then the probability of success is given by the ration n/N. Example: Probability of drawing a red

card from a pack of cards = 26/52 = 0.5.

Probability

Measures the likeliness of an outcome of an experiment.

The probability of an event is the proportion of the time that events of the same kind will occur in the long run.

Is a number between 0 and 1, with 0 and 1 included.

Probability of an event A is the ratio of favorable outcomes to event A to the total possible outcomes of that experiment.

Some Definitions

• An experiment is the process by which an observation or measurement is made.

• When an experiment is preformed it can result in one or more outcomes which are called events.

• A sample space associated with an experiment is the set containing all possible outcomes of that experiment. It is denoted by S.

• Elements of the sample space are called sample points.

Sample spaces

Can be Finite (toss a coin) Countable (toss a coin till the first head

appears) Discrete (toss a coin) Continuous (weight of a coin)

Example

• A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, two of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available.– List the sample space for this experiment.– Let A denote the event that the order is filled

with two nondefective terminals. List the sample points in A.

Venn Diagram

Graphical representation of sets. A’ is complement of event A.

Probability of an Event

POSTULATE 1: The probability of an event is a nonnegative real number that is P(A)>=0 for any subset A of S.

POSTULATE 2: P(S) = 1. POSTULATE 3: If A1, A2, ….is a finite

or infinite sequence of mutually exclusive events of S then P(A1 U A2 U…..)=P(A1 )+P(A2 )+……..

Mutually exclusive events

Two events A and B are said to be mutually exclusive if there are no elements common to the two sets. The intersection of the two sets is empty.

Rules of Probability

If A and A’ are complementary events in a sample space S, then P(A’)=1-P(A). Thus P(S) = 1.

P(Φ) = 0, probability of an empty set = 0.

If A and B are events in a sample space S and A is subset B the P(A) < =P(B).

If A and B are any two events in a sample space then P(A U B) = P(A)+P(B) – P(A B).

Conditional Probability

The conditional probability of an event A, given that an event B has occurred, is equal to

Provided P(B) > 0. Theorem: If A and B are any two

events in a sample space S and P(A)≠0, then

)(

)()|(

BP

BAPBAP

)|().()( BAPBPBAP

Independent Events

Two events A and B are independent if and only if

If A and B are independent then A and B’ are also independent.

)().()( BPAPBAP

Example

A coin is tossed three times. If A is the event that a head occurs on each of the first two tosses, B is the event that a tail occurs on the thrid toss and C is the event that exactly two tails occur in the three tosses, show that Events A and B are independent. Events B and C are dependent.

Bayes Theorem

If B1, B2, …., and Bk constitute a partition of the sample space S and P(Bi) ≠0, for i=1, 2, …., k, then for any event A in S such that P(A) ≠0

kr

BAPBP

BAPBPABP k

iii

rrr

,....,2,1

)|().(

)|().()|(

1

Reliability of a product

The reliability of a product is the probability that it will function within specified limits for a specific period of time under specified environmental conditions.

The reliability of a series system consisting of n independent components is given by

The reliability of a parallel system consisting of n independent components is given by Ri is the reliability of the ith component.

n

iis RR

1

n

iiP RR

1

)1(1