Dill and Bromberg
I.
Chapter 1
PROBABILITY
Principles of Probability
(pp 2-5)
Term
Definition
Specifics
Probability
pA = nA/N where nA = # of A outcomes
and N = total # all outcomes
0  pA 1
If outcome A occurs, outcome B cannot
If pA = 1, then deterministic
If pA = 0, then event never occurs
Mutually exclusive
events
Collectively
exhaustive
Independent
Multiplicity
Addition Rule
Multiplication
Rule
Prob of outcome
not happening
Prob of something
happening
Set of all possible outcomes
Uncorrelated or independent outcomes
Total number of possible outcomes
W = nAnB nC
If outcomes are mutually exclusive, then the
prob. of observing one outcome or another or
another is p =  pi
If outcomes are mutually exclusive and
collectively exhaustive, then N =  ni and p =
1
If outcomes are independent, then the prob of
observing two specific outcomes is p =  pi
If pA is prob that outcome A happens, then the
prob that outcome A does not happen is
(1-pA)
1-p(not A and not B) = 1-(1-pA) (1-pB)
Text Example
Heads or tails
Number on one die
(H, T)
(1, 2, 3, 4, 5, 6)
Coin flips, die rolls
Fig 1.1, Ex 1.3
Roll die once and get 1 or 4.
p = 1/6 + 1/6 = 1/3
Ex 1.1
Roll die twice and get 1 and 4.
p = 1/6 x 1/6 = 1/36
Ex 1.2, 1.3, 1.4
Ex 1.5
Ex 1.6
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II.
EVENTS
(pp 5-13)
The probability of some situations cannot be answered by the addition or multiplication rules alone, but reformulating the situation in
terms of composite events allows it to be answered by these rules.
Term
Definition
Elementary
Composite
A single event
Set of elementary events
Correlated or
Conditional
An event is conditional on the outcome
of an earlier event
If events are independent, then p(BA)
= p(B) and p(BA)p(A) = p(B) p(A) =
multiplication rule above; this is a priori
probability.
If events are conditional, then a
posteriori probability.
Correlated or
Conditional
If events are mutually exclusive, then
p(AB) = 0 and prob = addition rule
above
Specifics
Text Examples
When events are independent AND
composite events are mutually exclusive
Conditional prob: p(B(second
event)A(first event)) = p(BA)
Ex 1.7
Ex 1.8
Joint prob: p(AB) = prob that both events,
A and B, will occur
General Multiplication Rule (Bayes Rule):
p(AB) = p(BA)p(A) = p(AB)p(B)
Ex 1.9, T 1.1
p.4 is a special case
Holds for independent events and also
conditional events. p(AB) = p(AB)
General Addition Rule: p(AB) = p(A) +
p(B) - p(AB)
Ex 1.10
p. 3 is a special case
If events are independent, then p(AB)
= p(A)p(B)
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Term
Definition
Specifics
Correlated or
Conditional
If g = 1, events A and B are indep and
not correlated.
If g > 1, events A and B are positively
correlated.
If g < 1, events A and B are negatively
correlated.
If g = 0 and A occurs, B will not.
Counting events; concerned with the
composition of events not the sequence.
Degree of Correlation = g =
p(AB)/p(A)p(B) = conditional prob of B
divided by unconditional prob of B alone
Combinatorics
(distinguishable
objects)
Text Examples
W = N! = # permutations or different
sequences for N distinguishable objects.
Eqn 1.2
Ex 1.11
W = N!/( ni!) for indistinguishable
objects.
Eqn 1.18
Ex 1.13
Ex 1.12
Central to Thermodynamics
Entropy is the tendency of matter to
disorder
Combinatorics
(indistinguishable
objects)
Ex 1.14, Ex 1.15
Ex 1.16, Ex 1.17
Ex. 1.17
Consider n particles placed in M boxes. Or consider n particles with M-1 moveable box walls (Fig 1.3). How many permutations are
there for arranging n particles and M-1 walls? W(n, M) = (M+n-1)!/(M-1)! n! Eqn 1.120
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III.
DISTRIBUTION FUNCTIONS
(pp 13-21)
Term
Definition
Distribution Function (DF)
DF describes collections of probabilities = p(i) and
p(i) = 1.
Probability Density
= p(x) where x refers to a continuous property;
p(x)dx = 1 if normalized
Binomial Distribution
Function (t = 2)
P(n,N) = probability that a series of N trials has n A
outcomes and (N-n) B outcomes in any order =
Binomial Distribution = P(n,N) =
pn(1-p)N-n N!/n! (N-n)! Eqn 1.28
Multinomial DF (t = t)
DF average value
Expectation value
DF std deviation (variance)
 P(n,N) = 1 over n = 0 to n = N
P(n1, n2,… nt, N) = p1n1, p2n2,p3n3… ptnt x (N!/ ni!)
 ni = N over I = 1 to I = t
Consider a property f(x) which depends on the
property x. The DF p(x) describes the distribution
of x values (assume continuous). So the average
value of the property f(x) depends on the
distribution of x according to Eqn 1.36
<f(x)> =  f(x) p(x)dx =  f(x) ψ(x)dx / ψ(x)dx
σ2 = measure of distribution width =
<(x - <x>)2> = <x2> - <x>2
Specifics
Text
Examples
Each indepen elementary event
has 2 mutually exclusive
outcomes
Eqn 1.31
<a f(x)> = a<f(x)>
<f(x) + g(x)> = <f((x)> +
<g(x)>
ψ(x) is the quantum mech
wavefunction that describes a
particle (atom, electron, etc).
Let mean value = <x> = a =
constant
Ex 1.20, 1.21,
1.22, 1.23
same
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