Probability.docx

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Theory of Probability

Submitted By

Vinayak S Bhustalimath

12MMF0023


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Probability

Probability is a measure or estimation of how likely it is that something will happen

or that a statement is true. Probabilities are given a value between 0 (0% chanceorwill not happen) and 1 (100% chance orwill happen). The higher the degree of

probability, the more likely the event is to happen.

A probability is a way of assigning every event a value between zero and one, with

the requirement that the event made up of all possible results (in our example, the

event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the

assignment of values must satisfy the requirement that if you look at a collection of

mutually exclusive events (events with no common results.

The probability of an eventA is written as P(A), p(A) or Pr(A).[17]This mathematical

definition of probability can extend to infinite sample spaces, and even uncountable

sample spaces, using the concept of a measure.

The opposite orcomplementof an eventA is the event [notA] (that is, the event

ofA not occurring); its probability is given by P(notA) = 1 - P(A). As an example, the

chance of not rolling a six on a six-sided die is 1 (chance of rolling a

six) .

If both eventsA and B occur on a single performance of an experiment, this is called

the intersection or joint probability ofA and B, denoted as .

Independent probability

If two events,A and B are independent then the joint probability is

for example, if two coins are flipped the chance of both being heads

is
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Mutual ly exclusiv e

If either eventA or event B or both events occur on a single performance of an

experiment this is called the union of the events A and B denoted as .

If two events are mutually exclusive then the probability of either occurring is

For example, the chance of rolling a 1 or 2 on a six-sided die

is

Not mutually exclusive

If the events are not mutually exclusive then

For example, when drawing a single card at random from a regular deck

of cards, the chance of getting a heart or a face card (J,Q,K) (or one that

is both) is , because of the 52 cards of a deck 13 are

hearts, 12 are face cards, and 3 are both: here the possibilities included in

the "3 that are both" are included in each of the "13 hearts" and the "12face cards" but should only be counted once.

Conditional probability

Conditional probabilityis the probability of some eventA, given the

occurrence of some other event B. Conditional probability is

written , and is read "the probability ofA, given B". It is defined

by

If then is formally undefined by this expression.

However, it is possible to define a conditional probability for some

zero-probability events using a -algebra of such events (such as

those arising from a continuous random variable).


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For example, in a bag of 2 red balls and 2 blue balls (4 balls in total),

the probability of taking a red ball is ; however, when taking a

second ball, the probability of it being either a red ball or a blue ball

depends on the ball previously taken, such as, if a red ball was taken,

the probability of picking a red ball again would be since only 1

red and 2 blue balls would have been remaining.

The Complement Rule

Probability versus Statistics

Probability is the field of study that makes statements about what will occur when a

sample is drawn from a known population. Statistics is the field of study that

describes how samples are to be obtained and how inferences are to be made about

unknown populations.


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Independent Events

Two events are independent if the occurrence or nonoccurrence of one event does

notchange the probability of the other event.

Multiplication Rule for Independent Events

General Multiplication Rule For all events (independent or not):

Conditional Probability (when ):

Two Events Occurring Together

)()()( BPAPBandAP

)|()()( ABPAPBandAP

)|()()( BAPBPBandAP

0)( BP

)(

)()|(

BP

BandAPBAP


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Either or Both of Two Events Occurring

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur at the same time. Mutually

Exclusive are Disjoint

If A and B are mutually exclusive, then P(A and B) = 0

Addition Rules

IfA and B are mutually exclusive, then

P(A or B) = P(A) + P(B).

IfA and B are not mutually exclusive, then

P(A or B) = P(A) + P(B)P(A and B).


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Multiplication Rule for Counting

If there are n possible outcomes for E1 and m possible outcomes for event E2, then

there are a total of n X m or nm possible outcomes for the series of events E1

followed by E2.

This rule extends to outcomes involving three, four or more series of events.

Tree Diagrams

Displays the outcomes of an experiment consisting of a sequence of activities. The

total number of branches equals the total number of outcomes. Each unique

outcome is represented by following a branch from start to finish.

Tree Diagrams with Probability

We can also label each branch of the tree with its respective probability. To obtain

the probability of the events, we can multiply the probabilities as we work down a

particular branch.

Example

Suppose there are five balls. Three are red and two are blue. We will select a ball,

note the color, and, without replacing the first ball, select a second ball.


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There are four possible outcomes:

1. Red, Red 2. Red, Blue 3. Blue, Red 4. Blue, Blue

We can find the probabilities of the outcomes by using the multiplication rule for

dependent events.

Set Operations

Union

Intersection

Complement

Properties

Commutation

Associativity

Distribution

De Morgans Rule

A B

A B

CA

A B B A

A B C A B C

A B C A B A C

C C CA B A B

A B

CA

S

A B


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Conditional Probability

Conditional Probability of event A given that event B has occurred

If B1, B2,,Bn a part i t ionof S, then

(Law of Total Probability)

|

P A BP A B

P B

1 1| ...

| j j

P A P A B P B

P A B P B

A B

CA

S

A B

B1

B3

B2

A


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Questions:

1. Define Probability.

2. Define Independent probability.

3. Define Conditional Probability

4. Define conditional probability.

5. What is a tree diagram? Explain with example.

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Probability.docx