lecture 7-8
PROBABILITY :
denoted by P and mathematically we can define as
P = number of favourable events /number of exhaustive events
for example
1. toss a coin
* probability of occurrence of head probability
P(H) = 1/2
Probability of occurance of tail is :
P(T) =1/2
RANGE OF PROBABILITY :
0<P<1
* If P is zero it means event will definitely not occurs .
example throwing a dies and find the probability of getting number less than or equal to 6
P( number less than or equal to 6 )= 6/6 = 1
Types of Probability :
CLASSICAL PROBABILITY :
In the classical probability we are assign the probability of all possible outcomes.
Pi = 1/w
this is very useful working hypothesis in the occurrence of any information.
for example (1) toss a coin .(2)throwing a dice .
STATISTICAL PROBABILITY :
statistical probability also known as relative frequency approach .It is related with experiment methods of assigning probabilities to events by measuring relative frequency of occurrence .
If the random process is repeated N times and an event ” A” occurs Na times then
P(A) = Lim Napproaches to infinity NA/N
If there are flips of coin that probability of head occurrence is
P(Head ) = Lim Napproaches to infinity Nhead/N flips
example
if it also coin 7000 times and head occurs 4000 times then the probability of occurrence of head according to relative frequences .
=4000/7000
=.57
Axioms of probability :
In the statistical mechanics axioms of probability theory are :
(1) .probability are positive or zero .
(2).the probability are less than or equal to zero .
(3).In case of compound events i +j means that either event i occurs or event j occurs or both at same time.
mutually exclusive events probability of event i+j is
p(i+j) =Pi +Pj
PERMUTATION AND COMBINATION :
INTRODUCTION
When we use the classical probability it is necessary to work out the number of different events that can occurs .
Counting the number of ways which is variants events can happen leads to the concept of arguments” permutations and combinations” .permutation means ”arrangement” and the word combination mean ”formation of groups ” .
| PERMUTATION | COMBINATION |
| Permutation is the arrangement of items in which order matters . | Combination is the selection of items in order doesn’t matte . |
| Formula: nPr = n!/(n-r)! | Formula: nCr =n!/r! (n-r)! |
| In permutation repetation is not allowed. | In combination repetation is allowed . |
| It means you are going to arrange ‘’r ‘’ things out og ‘’n’’ things So, one can say that permutation is a number of ways to arrange things . | It means you are going to select ‘’s’’ things So one can say that combination is the number of ways to choose or select things . |
In how many ways you can arrage letteers in the wors ”LARGE ”?
*IF REPETITION IS NOT ALLOWED :-
= 5X4X3X2X1
= 5!
=120 ways .
*IF REPITITION IS ALLOWED :-
= 5X5X5X5X5
=(5)5 Ways .
PRINCIPLE OF EQUAL PRIOR PROBABILITY :
statement :
The principal of assuming equal probability for events which are equally likely is known as principal of priopri probability.
explanation :
when we toss a coin we are clear in the mind that either it is a head or tail .similarly, by throwing a dice show that one of its 6 faces upward the same way.
In the same way we have an open box divided into equal size compartments and a small particle is thrown from a large distance distance in such a way that it must fall in the of two compartments marked ”A ”is equal to probability that it may fall into the compartment ”B” .
Note :-
Principal of equal to a probability will not hold well if in above example the two compartments are of unequal sized.
Conditions :-
1. events should be mutually exclusive events.
2. should be equally likely .
foundation of statistical mechanics statistical mechanics :-
Let us consider a system of an ideal gas and closed in a cylinder than according to theory of gases their exist 10 23 number of molecules per unit volume .Now if we want to describe such system according to write 1023 number of equation and then solve them which is very difficult approach for us .Therfore get the answer of problem we have to use the technique of statistical mechanics thus we define STATISTICAL MECHANICS :
definition
The branch of physics with deals with the statistical predictions of properties of matter in bulk form from a knowledge of the properties of constituents i.e atoms are molecules and force between them the statistical prediction sometimes restricted to the treatment of system in equilibrium.
statistical physics = is the study of microscopic properties of system in terms of its microscopic properties.
| imp | STATISTIACAL MECHANICS | |||
| CLASSICAL STATISTICAL MECHANICS | QUANTUM STATISTICAL MECHANICS | |||
| Based on Maxwell-Boltzman statistics . | Fermi –Dirac Statistics. | Boes –Einstein Statistics . | ||
| 1.Particles are identical and distinguishable. | 1.Particles are identical and indistinguishable. | 1. Particles identical and in distinguishable. | ||
| 2. Particles have any spin or any spin as spin doesn’t matter . | 2.Half integral spin (1/2,3/2,5/2)h For example : electron ,protrons . | 2.Particles for example photon. | ||
| 3. Energy will become zero at absolute zero (T=0) for example gas molecule | 3.Energy will not equal to absolute zero temperature . | 3.Energy will become zero at absolute zero temperature Or integral multiple of | ||
| 4. There is no exception on particles more than two particles occupy the same space | 4.There is restriction on particles .As particle follow Paulis Exclusion Principle . | 4.There is also number of restrictions on particles. | ||
| 5.Localized particles ψ don’t overlap . | 5.Wave function overlap and total ψ is asymmetric. | 5.Wave function overlap total ψ symmetric | ||
