Thermodynamics and Statistical
Mechanics
Probabilities
Thermo & Stat Mech Spring 2006 Class 16
1
Pair of Dice
For one die, the probability of any face coming
up is the same, 1/6. Therefore, it is equally
probable that any number from one to six will
come up.
For two dice, what is the probability that the
total will come up 2, 3, 4, etc up to 12?
Thermo & Stat Mech - Spring 2006
Class 16
2
Probability
To calculate the probability of a particular
outcome, count the number of all possible
results. Then count the number that give the
desired outcome. The probability of the desired
outcome is equal to the number that gives the
desired outcome divided by the total number of
outcomes. Hence, 1/6 for one die.
Thermo & Stat Mech - Spring 2006
Class 16
3
Pair of Dice
List all possible outcomes (36) for a pair of dice.
Total
Combinations
How Many
2
1+1
1
3
1+2, 2+1
2
4
1+3, 3+1, 2+2
3
5
1+4, 4+1, 2+3, 3+2
4
6
1+5, 5+1, 2+4, 4+2, 3+3
5
Thermo & Stat Mech - Spring 2006
Class 16
4
Pair of Dice
Total
7
8
9
10
11
12
Combinations
How Many
1+6, 6+1, 2+5, 5+2, 3+4, 4+3
6
2+6, 6+2, 3+5, 5+3, 4+4
5
3+6, 6+3, 4+5, 5+4
4
4+6, 6+4, 5+5
3
5+6, 6+5
2
6+6
1
Sum = 36
Thermo & Stat Mech - Spring 2006
Class 16
5
Probabilities for Two Dice
Total 2
1
Prob.
36
% 2.8
3
2
36
5.6
4
3
36
8.3
5
4
36
11
6
5
36
14
7
6
36
17
8
5
36
14
Thermo & Stat Mech - Spring 2006
Class 16
9 10 11 12
4 3 2 1
36 36 36 36
11 8.3 5.6 2.8
6
Probabilities for Two Dice
D ice
Pr obability
0 .2
0 .1 5
0 .1
0 .0 5
0
2
3
4
5
6
7
8
9
10
11
12
Nu m b e r
Thermo & Stat Mech - Spring 2006
Class 16
7
Microstates and Macrostates
Each possible outcome is called a “microstate”.
The combination of all microstates that give
the same number of spots is called a
“macrostate”.
The macrostate that contains the most
microstates is the most probable to occur.
Thermo & Stat Mech - Spring 2006
Class 16
8
Combining Probabilities
If a given outcome can be reached in two (or
more) mutually exclusive ways whose
probabilities are pA and pB, then the probability
of that outcome is: pA + pB.
This is the probability of having either A or B.
Thermo & Stat Mech - Spring 2006
Class 16
9
Combining Probabilities
If a given outcome represents the combination
of two independent events, whose individual
probabilities are pA and pB, then the probability
of that outcome is: pA × pB.
This is the probability of having both A and B.
Thermo & Stat Mech - Spring 2006
Class 16
10
Example
Paint two faces of a die red. When the die is
thrown, what is the probability of a red face
coming up?
1 1 1
p  
6 6 3
Thermo & Stat Mech - Spring 2006
Class 16
11
Another Example
Throw two normal dice. What is the probability
of two sixes coming up?
1 1 1
p ( 2)   
6 6 36
Thermo & Stat Mech - Spring 2006
Class 16
12
Complications
p is the probability of success. (1/6 for one die)
q is the probability of failure. (5/6 for one die)
p + q = 1,
or q = 1 – p
When two dice are thrown, what is the
probability of getting only one six?
Thermo & Stat Mech - Spring 2006
Class 16
13
Complications
Probability of the six on the first die and not
the second is:
1 5 5
pq   
6 6 36
Probability of the six on the second die and not
the first is the same, so:
10 5
p (1)  2 pq 

36 18
Thermo & Stat Mech - Spring 2006
Class 16
14
Simplification
Probability of no sixes coming up is:
5 5 25
p (0)  qq   
6 6 36
The sum of all three probabilities is:
p(2) + p(1) + p(0) = 1
Thermo & Stat Mech - Spring 2006
Class 16
15
Simplification
p(2) + p(1) + p(0) = 1
p² + 2pq + q² =1
(p + q)² = 1
The exponent is the number of dice (or tries).
Is this general?
Thermo & Stat Mech - Spring 2006
Class 16
16
Three Dice
(p + q)³ = 1
p³ + 3p²q + 3pq² + q³ = 1
p(3) + p(2) + p(1) + p(0) = 1
It works! It must be general!
(p + q)N = 1
Thermo & Stat Mech - Spring 2006
Class 16
17
Binomial Distribution
Probability of n successes in N attempts
(p + q)N = 1
N!
n N n
P ( n) 
p q
n!( N  n)!
where, q = 1 – p.
Thermo & Stat Mech - Spring 2006
Class 16
18
Thermodynamic Probability
The term with all the factorials in the previous
equation is the number of microstates that will
lead to the particular macrostate. It is called the
“thermodynamic probability”, wn.
N!
wn 
n!( N  n)!
Thermo & Stat Mech - Spring 2006
Class 16
19
Microstates
The total number of microstates is:
  w
wn
True probabilit y P(n) 

For a very large number of particles
  wmax
Thermo & Stat Mech - Spring 2006
Class 16
20
Mean of Binomial Distribution
n   P ( n) n
n
where
N!
n N n
P ( n) 
p q
n!( N  n)!

Notice : p P(n)  P(n)n
p
Thermo & Stat Mech - Spring 2006
Class 16
21
Mean of Binomial Distribution

n   P ( n) n   p P ( n)
p
n
n


N
n  p  P ( n)  p ( p  q )
p n
p
n  pN ( p  q )
N 1
 pN (1)
N 1
n  pN
Thermo & Stat Mech - Spring 2006
Class 16
22
Standard Deviation (s)
n  n 
s
2
s  n  n    P (n)n  n 
2
2
2
n
n  n 
2
 n  2n n  n  n  2n n  n
2
2
s n n
2
2
2
2
2
Thermo & Stat Mech - Spring 2006
Class 16
23
Standard Deviation
2
 
n   P ( n) n   p   P ( n)
n
 p  n
    
 
2
N
N 1
n   p  p ( p  q )   p  pN ( p  q )
 p  p 
 p 
2
2
n  pN ( p  q )
2
N 1
 pN ( N  1)( p  q )
N 2

n 2  pN 1  pN  p   pN q  pN 
Thermo & Stat Mech - Spring 2006
Class 16
24
Standard Deviation
s n n
2
2
2
s 2  pN q  pN   ( pN ) 2
s 2  Npq  ( pN ) 2  ( pN ) 2  Npq
s  Npq
Thermo & Stat Mech - Spring 2006
Class 16
25
For a Binomial Distribution
n  pN
s  Npq
s
q

n
Np
Thermo & Stat Mech - Spring 2006
Class 16
26
Coins
Toss 6 coins. Probability of n heads:
n
6 n
N!
6!
1 1
n N n
P ( n) 
p q

   
n!( N  n)!
n!(6  n)!  2   2 
6!
1
P ( n) 
 
n!(6  n)!  2 
6
Thermo & Stat Mech - Spring 2006
Class 16
27
For Six Coins
Binomial Distribution
0.35
0.3
Probabilty
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
Successes
Thermo & Stat Mech - Spring 2006
Class 16
28
For 100 Coins
Binomial Distribution
0.09
0.08
Probabilty
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
6
12
18
24
30
36
42
48
54
60
66
72
78
84
90
96
Successes
Thermo & Stat Mech - Spring 2006
Class 16
29
For 1000 Coins
Binomial Distribution
0.03
0.02
0.015
0.01
0.005
960
900
840
780
720
660
600
540
480
420
360
300
240
180
120
60
0
0
Probabilty
0.025
Successes
Thermo & Stat Mech - Spring 2006
Class 16
30
Multiple Outcomes
N!
N!
w

N1! N 2 ! N 3!     N i !
N
i
N
i
Thermo & Stat Mech - Spring 2006
Class 16
31
Stirling’s Approximation
For large N : ln N ! N ln N  N
 N! 
  ln N ! ln  N i !  ln N ! ln N i !
ln w  ln 
i
  Ni! 


ln w  N ln N  N    ( N i ln N i )   N i 
i
 i

ln w  N ln N   ( N i ln N i )
i
Thermo & Stat Mech - Spring 2006
Class 16
32
Number Expected
Toss 6 coins N times. Probability of n heads:
n
N!
6!
1 1
n N n
P ( n) 
p q

   
n!( N  n)!
n!(6  n)!  2   2 
6 n
6
6!
1
P ( n) 
 
n!(6  n)!  2 
Number of times n heads is expected is:
n = N P(n)
Thermo & Stat Mech - Spring 2006
Class 16
33