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# Statistical Reasoning

In the logic based approaches described, we have assumed that everything is either believed false or believed true.

Statistical Reasoning :

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In the logic based approaches described, we have assumed that everything is either believed false or believed true.

However, it is often useful to represent the fact that we believe such that something is probably true, or true with probability (say) 0.65.

This is useful for dealing with problems where there is randomness and unpredictability (such as in games of chance) and also for dealing with problems where we could, if we had sufficient information, work out exactly what is true.

To do all this in a principled way requires techniques for probabilistic reasoning. In this section, the Bayesian Probability Theory is first described and then discussed how uncertainties are treated.

Recall glossary of terms

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Ō¢Ā Probabilities :

Usually, are descriptions of the likelihood of some event occurring (ranging from 0 to 1).

Ō¢Ā Event :

One or more outcomes of a probability experiment .

Ō¢Ā                Probability Experiment :

Process which leads to well-defined results call outcomes.

Ō¢Ā                Sample Space :

Set of all possible outcomes of a probability experiment.

Ō¢Ā  Independent Events :

Two events, E1 and E2, are independent if the fact that E1 occurs does not affect the probability of E2 occurring.

Ō¢Ā Mutually Exclusive Events :

Events E1, E2, ..., En are said to be mutually exclusive if the occurrence of any one of them automatically implies the non-occurrence of the remaining n ŌłÆ 1 events.

Ō¢Ā                Disjoint Events :

Another name for mutually exclusive events.

Ō¢Ā Classical Probability :

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Also called a priori theory of probability.

The probability of event A = no of possible outcomes f divided by the total no of possible outcomes n ; ie., P(A) = f / n.

Assumption: All possible outcomes are equal likely.

Ō¢Ā                Empirical Probability :

Determined analytically, using knowledge about the nature of the experiment rather than through actual experimentation.

Ō¢Ā                Conditional Probability :

The probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and read as "the probability of A, given B ".

Ō¢Ā                Joint probability :

The probability of two events in conjunction. It is the probability of both events together. The joint probability of A and B is written P(A Ōł®

; also written as P(A, B).

Marginal Probability :

The probability of one event, regardless of the other event. The marginal probability of A is written P(A), and the marginal probability of B is written P(B).

Examples

Ō¢Ā Example 1

Sample Space - Rolling two dice

The sums can be { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }.

Note that each of these are not equally likely. The only way to get a sum 2 is to roll a 1 on both dice, but can get a sum 4 by rolling out comes as (1,3), (2,2), or (3,1).

Table below illustrates a sample space for the sum obtain. Classical Probability

Table below illustrates frequency and distribution for the above sums. The classical probability is the relative frequency of each event.

Classical probability P(E) = n(E) / n(S); P(6) = 5 / 36, P(8) = 5 / 36

Empirical Probability

The empirical probability of an event is the relative frequency of a frequency distribution based upon observation P(E) = f / n

Ō¢Ā       Example 2

Mutually Exclusive Events (disjoint) : means nothing in common

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

If two   events are mutually exclusive,

then probability of both occurring at same time is P(A and B) = 0

If two events are mutually exclusive ,

then the probability of either occurring is P(A or B) = P(A) + P(B)

Given P(A)= 0.20, P(B)= 0.70, where A and B are disjoint

then P(A and B) = 0

The table below indicates intersections ie "and" of each pair of events. "Marginal" means total; the values in bold means given; the rest of the values are obtained by addition and subtraction. Non-Mutually Exclusive Events

The non-mutually exclusive events have some overlap.

When P(A) and P(B) are added, the probability of the intersection (ie. "and" ) is added twice, so subtract once.

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

Given :  P(A) = 0.20,               P(B) = 0.70,  P(A and B) = 0.15 Ō¢Ā                Example 3

Factorial , Permutations and Combinations

Factorial

The factorial of an integer n Ōēź 0 is written as  n! .

n! = n ├Ś n-1 ├Ś . . . ├Ś 2 ├Ś 1. and    in particular,    0! = 1.

It is, the number of permutations of n distinct objects;

e.g., no of ways to arrange 5 letters A, B, C, D and E   into a word is 5!

5!      = 5 x 4  x    3  x 2 x 1 = 120

N!      = (N) x (N-1) x (N-2) x . . . x (1)

n!      = n (n - 1)! ,          0! = 1

Permutation

The permutation is arranging elements (objects or symbols) into distinguishable sequences. The ordering of the elements is important. Each unique ordering is a permutation.

Number of permutations of ŌĆ× n ŌĆ¤ different things taken ŌĆ× r ŌĆ¤ at a time is given by (for convenience in writing, here after the symbol Pnr is written as nPr or P(n,r) )

Example 1

Consider a total of 10 elements, say integers {1, 2, ..., 10}. A permutation of 3 elements from this set is (5, 3, 4). Here n = 10 and r = 3.

The number of such unique sequences are calculated as P(10,3) = 720.

Example 2

Find the number of ways to arrange the three letters in the word

CAT in to two-letter groups like CA or AC and no repeated letters.

This means permutations are of size r = 2 taken from a set of size n = 3. so P(n, r) = P(3,2) = 6.

The ways are listed as CA CT AC AT TC TA.

Similarly, permutations of size r = 4, taken from a set of size n = 10, Combinations

Combination means selection of elements (objects or symbols). The ordering of the elements has no importance.

Number of Combination of ŌĆ× n ŌĆ¤ different things, taken ŌĆ× r ŌĆ¤ at a time is (for convenience in writing, here after the symbol Cnr is written as nCr or C(n,r) )

Example

Find the number of combinations of size 2 without repeated letters that can be made from the three letters in the word CAT, order doesn't matter;

This means combinations of size r =2 taken from a set of size n = 3, Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
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