Page 331 - Numerical Methods for Chemical Engineering

320 7 Probability theory and stochastic simulation

way as to avoid inconsistencies. For example, the sum of the probabilities that E occurs and
that E does not occur must always equal 1. Similarly, we define probabilities such that they
satisfy the rules of joint and conditional probabilities outlined below.
We now use this probabilistic (frequentist) interpretation to compute [P n ]. Each chain in
the system, being linear, has on average one unreacted acid group and one unreacted base
group. As the acid and base concentrations are equal, the total number of chains is
∞

(7.4)
[P n ] = [A] = (1 − p)[A] 0
n=1
What fraction of this total number of chains has a length equal to n? Or, equivalently, what
is the probability that a randomly selected unreacted acid group is attached to a chain with
exactly n monomer units? If this probability is P(A is attached to chain of n units), then
[P n ] = (1 − p)[A] 0 × P(A is attached to chain of n units) (7.5)

To compute this quantity, we use the rules of probability theory, for which we need a few
basic definitions.
Probability theory deals with events. Here, the event of interest is the observation that
a randomly-selected chain has exactly n monomer units. Let us say that we have selected
at random an unreacted acid end group that must lie at a chain end. If we march down the
chain, we find that to achieve a length of exactly n units, the first (n − 1) encountered acid
th
groups must have reacted and the n group must not have reacted. Thus, the event that our
chain has exactly n units can be related to a sequence of simpler events – whether each
of the acid groups encountered along the chain has reacted or not. We now compute the
probability of the composite event (the chain contains n units) from the probabilities of
these simpler events.
th
For each j = 1, 2,..., N, let R j be the event that the j acid group along the chain has
reacted, and U j be the event that it is unreacted. Since these are the only two possibilities,
their probabilities must sum to 1,
P(R j ) + P(U j ) = 1 (7.6)

Our composite event, (A is attached to chain of n units), is equivalent to saying that the
following sequence of events occurs, R 1 , R 2 ,..., R n−1 , U n so that P(A is attached to chain
of n units) is equal to the joint probability

P(A is attached to chain of n units) = P(R 1 ∩ R 2 ∩···∩ R n−1 ∩ U n ) (7.7)
∩ is the symbol for intersection, and here signifies that the events on both sides of the sign
occur.

Defining joint and conditional probabilities

The joint probability of two events E 1 and E 2 is the probability P(E 1 ∩ E 2 ), also written
P(E 1 , E 2 ), that both occur. If P(E 1 ) is the probability that E 1 occurs,

P(E 1 ∩ E 2 ) = P(E 1 )P(E 2 | E 1 ) (7.8)