Concrete mathematics : a foundation for computer science

394 DISCRETE PROBABILITY
In the special case p = i we can interpret these formulas in a particularly
simple way. Given a pattern A of m heads and tails, let
A:A = fIkpl [Ack’ =A(kj] .
k=l
(8.76)
We can easily find the binary representation of this number by placing a ‘1’
under each position such that the string matches itself perfectly when it is
superimposed on a copy of itself that has been shifted to start in this position:
A = HTHTHHTHTH
A:A=(1000010101)2=-512+16+4+l =533
HTHTHHTHTH J
HTHTHHTHTH
HTHTHHTHTH
HTHTHHTHTH
HTHTHHTHTH
HTHTHHTH'TH J
HTHTHHTHTH
HTHTHHTHTH J
HTHTHHTHTH
HTHTHHTHTH J
Equation (8.74) now tells us that the expected number of flips until pattern A
appears is exactly 2(A:A), if we use a fair coin, because &kj = Ik when
p=q=$. This result, first discovered by the Soviet mathematician A. D. “Chem bol’she
Solov’ev in 1966 [271], seems paradoxical at first glance: Patterns with no periodov u nasheg0
s/ova, tern pozzhe
self-overlaps occur sooner th,an overlapping patterns do! It takes almost twice on0 poMl~ets%”
as long to encounter HHHHH as it does to encounter HHHHT or THHHH. -A. D. Solov’ev
Now let’s consider an amusing game that was invented by (of all people)
Walter Penney [231] in 196!3. Alice and Bill flip a coin until either HHT or
HTT occurs; Alice wins if the pattern HHT comes first, Bill wins if HTT comes
first. This game-now called “Penney ante” -certainly seems to be fair, if
played with a fair coin, because both patterns HHT and HTT have the same
characteristics if we look at them in isolation: The probability generating
function for the waiting tim’e until HHT first occurs is
G(z) =
z3
z3 - 8(2- 1) ’
and the same is true for HTT. Therefore neither Alice nor Bill has an advan-
Of w not! Who
could they have an
tage, if they play solitaire. advantage over?