Concrete mathematics : a foundation for computer science

402 DISCRETE PROBABILITY
Let sj be the probability that we are searching for the jth key that was
inserted into the table. Then
Pr(w) = sk/mn (8.87)
if w is the event (8.86). (Some applications search most often for the items
that were inserted first, or for the items that were inserted last, so we will not
assume that each Sj = l/n.) Notice that ,&l Pr(w) = Et=, sk = 1, hence
(8.87) defines a legal probability distribution.
The number of probes P in a successful search is p if key K was the pth
key to be inserted into its hst. Therefore
P = [h, = h-k] + [hz = hkl + . . . + [hk =hkl ;
or, if we let Xj be the random variable [hj = hk], we have
P = x1 +& + “‘+xk. (8.88)
Suppose, for example, that we have m = 10 and n = 16, and that the hash
values have the following “random” pattern: Where have I seen
that pattern before?
(h-l,..., h,6)=3 141592653589793;
(Pl,. . *, P,~)=1112111122312133.
The number of probes Pj needed to find the jth key is shown below hi.
Equation (8.88) represents P as a sum of random variables, but we can’t
simply calculate EP as EX, $-. . .+EXk because the quantity k itself is a random
variable. What is the probability generating function for P? To answer this
question we should digress #a moment to talk about conditional probability. Equation (8.43) was
If A and B are events in a probability space, we say that the conditional a1so a momentary
digression.
probability of A, given B, is
F’r(cu g A n B)
Pr(wEAIwEB) = - Pr(wCB) ’
For example, if X and Y are random variables, the conditional probability of
the event X = x, given that Y = y, is
Pr(X=x and Y=y)
Pr(X=xlY=y) = - Pr(Y=y) ’
(8.90)
For any fixed y in the range of Y, the sum of these conditional probabilities
over all x in the range of X is Pr(Y =y)/Pr(Y =y) = 1; therefore (8.90)
defines a probability distribution, and we can define a new random variable
‘X/y’ such that Pr(Xly =x) = Pr(X =x 1 Y =y).