18^th July lecture:

There are 3 Methods of estimation

1.    Ratios chance ( which vary from 0 to 1  eg.1/10)

2.    Percentage chance ( which vary from  0 to 100  eg. 50% )

3.    Probability (which vary from 0 to 1 eg. 0.1)

Probability Method of estimation

3 approaches to calculate Probability were showed

1.    A prior ( prior information  For Eg. for example, suppose p is the proportion of voters who will vote for the politician named Obama in a future election)

The probability that an event will reflect established beliefs about the event before the arrival of new evidence or information. Prior probabilities are the original probabilities of an outcome, which be will updated with new information to create posterior probabilities.

2.    Empirical approach ( information collected )

Also known as relative frequency, or experimental probability, is the ratio of the number of outcomes in which a specified event occurs to the total number of trials,[1][2] not in a theoretical sample space but in an actual experiment. In a more general sense, empirical probability estimates probabilities from experience and observation.

3.    Subjectivity ( Intuition )

A probability derived from an individual's personal judgment, understanding and experience  about whether a specific outcome is likely to occur. Subjective probabilities contain no formal calculations. This can be used to capitalize on background of experienced workers and managers in decision making.

It’s a way of tapping a persons knowledge to forecast the occurrence of an event.

Sample Place:

In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment.

For example: if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

Venn Diagrams

The sample space and an event may be represented on a Venn diagram.

For the experiment of tossing a fair coin, the possible outcomes are head and tail.  So, the following Venn diagram represents the experiment's sample space.

If A is the event 'a head falls', then we can use the following Venn diagram to represent it.

Types of VENN diagrams:

Bayes Rule

It was developed by Thomas Bayes.

It is a formula that extends the use of law of conditional probabilities to allow revision of original probabilities with new information.

Probability: Types of Events

When we say "Event" we mean one (or more) outcomes.

Independent Events

Events can be "Independent", meaning each event is not affected by any other events.

INDEPENDENT EVENTS X AND Y == P(X|Y)=P(X) AND P(Y|X)=P(Y)

Example: left handedness is probably independent of possession of a credit card. Whether a person wears glasses or not is probably independent of the brand of milk.

You toss a coin three times and it comes up "Heads" each time ... what is the chance that the next toss will also be a "Head"?

The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin.

Dependent Events

But some events can be "dependent" ... which means they can be affected by previous events .

Example: Drawing 2 Cards from a Deck

After taking one card from the deck there are less cards available, so the probabilities change!

Mutually Exclusive

It is either one or the other, but not both

Examples:

• Turning left or right are Mutually Exclusive (you can't do both at the same time)
• Heads and Tails are Mutually Exclusive
• Kings and Aces are Mutually Exclusive
• An office building is for sale and two different potential buyers have placed bids on the building, its not possible for both buys to purchase the building; therefore, the event of buyer A purchasing the building is mutually exclusive with the event of buyer B purchasing the building

The probability of two mutually exclusive events occurring at the same time is ZERO

What isn't Mutually Exclusive.

• Kings and Hearts are not Mutually Exclusive, because you can have a King of Hearts!

Then we further discussed LAWS OF PROBABILITY:

Addition law :

The Addition Law of Probability

If two events A and B are mutually exclusive then P(A∪B)=P(A)+P(B).

 This is the simplified version of the Addition Law. However, when A and B are not mutually exclusive, A∩B = ∅, it can be shown that a more general law applies: P(A∪B)=P(A)+P(B)−P(A∩B) Of course if A ∩ B = ∅ then, since P(∅) = 0 this general expression reduces to the simpler

Conditional Probability:

          A conditional probability is the probability that an event will occur, when    another event is known to occur or to have occurred

P(B|A)= P(A∩B) P(A)

or, equivalently P(A∩B)=P(B|A)P(A)

The Multiplication Law:

If A and B are independent events then P(A∩B)=P(A)P(B) In words

‘The probability of independent events A and B occurring is the product of the probabilities of the events occurring separately.’

We also did questions in the class related to the topics covered so as to enhance our concepts.

Written by: 

Priyanka Sudan

Group members:

Nishant Renjith

Pooja Shukla

Pranshu Agrawal

Prateek Jain

Priyanka Sudan