Concrete mathematics : a foundation for computer science

” ‘You really are an
automaton-a calculating
machine, ’
I cried. ‘There is
something positively
inhuman in you at
times.“’
-J. H. Watson (701
8.4 FLIPPING COINS 391
Now let’s try a more intricate experiment: We will flip coins until the
pattern THTTH is first obtained. The sum of winning positions is now
S = THTTH + HTHTTH + TTHTTH
+ HHTHTTH + HTTHTTH + THTHTTH + TTTHTTH + . ;
this sum is more difficult to describe than the previous one. If we go back to
the method by which we solved the domino problems in Chapter 7, we can
obtain a formula for S by considering it as a “finite state language” defined
by the following “automaton”:
The elementary events in the probability space are the sequences of H’s and
T’s that lead from state 0 to state 5. Suppose, for example, that we have
just seen THT; then we are in state 3. Flipping tails now takes us to state 4;
flipping heads in state 3 would take us to state 2 (not all the way back to
state 0, since the TH we’ve just seen may be followed by TTH).
In this formulation, we can let Sk be the sum of all sequences of H’s and
T’s that lead to state k: it follows that
so = l+SoH+SzH,
S1 = SoT+S,T+SqT,
S2 = S, H+ S3H,
S3 = S2T,
S4 = SST,
S5 = S4 H.
Now the sum S in our problem is S5; we can obtain it by solving these six
equations in the six unknowns SO, S1, . . . , Sg. Replacing H by pz and T by qz
gives generating functions where the coefficient of z” in Sk is the probability
that we are in state k after n flips.
In the same way, any diagram of transitions between states, where the
transition from state j to state k occurs with given probability pj,k, leads to
a set of simultaneous linear equations whose solutions are generating functions
for the state probabilities after n transitions have occurred. Systems
of this kind are called Markov processes, and the theory of their behavior is
intimately related to the theory of linear equations.