Business Statistics 18_07_2013
Things we discussed in 18th
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 e.g. 50% )
3. Probability (which vary
from 0 to 1 e.g. 0.1)
In class today we discussed and learned about Probability Method of estimation
3 approaches to calculate Probability
were showed
1. A prior ( prior information For E.g.for example, suppose p is the
proportion of voters who will vote for the politician named Smith 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 we will
update 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
about whether a specific outcome is likely to occur. Subjective probabilities
contain no formal calculations and only reflect the subject's opinions and past
experience
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:
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.
Example: If
you toss a coin three times and it comes up Heads each time,the chance of next
toss will be Head is simple again 50 %, Just like another toss of coin.
Dependent Events
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
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.
Bayes Theorem :
It is a extension to
conditional probability law.
The theorem expresses
how a subjective degree of belief should rationally change to account for
evidence.
Usually used to find
the cause of a given experimental result
P(Ai|B)=P(B|Ai)P(Ai)/∑j=ik(B|Aj)P(Aj)
Submitted By: Parth Mehta
Group Members:
Nikita Agarwal 2013171
Nimisha Agarwal 2013173
Nihal Moidu
Nimisha Agarwal 2013173
Nihal Moidu
Priyesh Bhadauriya 2013214
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