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 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
10
Example
Paint two faces of a die red. When the die is thrown, what is the probability of a red face coming up?
Thermo & Stat Mech - Spring 2006 Class 16 5
Probabilities for Two Dice
Total 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 5 4 3 2 1 Prob. 36 36 36 36 36 36 36 36 36 36 36 % 2.8 5.6 8.3 11 14 17 14 11 8.3 5.6 2.8
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.
1 1 1 p 6 6 3
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
6
Probabilities for Two Dice
D ice
0 .2
P r o b a b ilit y
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
பைடு நூலகம்
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 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