Concrete mathematics : a foundation for computer science

416 DISCRETE PROBABILITY
28 What is the average length of the coin-flipping game (8.78)
a given that Alice wins?
b given that Bill wins?
29 Alice, Bill, and Computer flip a fair coin until one of the respective
patterns A = HHTH, B := HTHH, or C = THHH appears for the first time.
(If only two of these patterns were involved, we know from (8.82) that A
would probably beat B, that B would probably beat C, and that C would
probably beat A; but all three patterns are simultaneously in the game.)
What are each player’s chances of winning?
30 The text considers three kinds of variances associated with successful
search in a hash table. Actually there are two more: We can consider the
average (over k) of the variances (over hl , . . . , h,) of P( hr , . . . , h,; k); and
we can consider the variance (over k) of the averages (over hl, . . . , h,,).
Evaluate these quantities.
31 An apple is located at vertex A of pentagon ABCDE, and a worm is
located two vertices away, at C. Every day the worm crawls with equal
probability to one of the two adjacent vertices. Thus after one day the
worm is at vertex B with probability i and at vertex D with probability i.
After two days, the worm might be back at C again, because it has no
memory of previous positions. When it reaches vertex A, it stops to dine.
a What are the mean and variance of the number of days until dinner?
b Let p be the probability that the number of days is 100 or more.
What does Chebyshev’s inequality say about p?
C What do the tail inequalities (exercise 12) tell us about p?
32 Alice and Bill are in t:he military, stationed in one of the five states
Kansas, Nebraska, Missouri, Oklahoma, or Colorado. Initially Alice is in
Nebraska and Bill is in Oklahoma. Every month each person is reassigned
to an adjacent state, each adjacent state being equally likely. (Here’s a
diagram of the adjacencies:
Schrtidinger’s worm.
The initial states are circled.) For example, Alice is restationed after the Definitely a finitefirst
month to Colorado., Kansas, or Missouri, each with probability l/3. state situation.
Find the mean and variance of the number of months it takes Alice and
Bill to find each other. (You may wish to enlist a computer’s help.)