Concrete mathematics : a foundation for computer science

(Use a calculator for
the numerical work
on this problem.)
33
34
Are the random variables X1 and X2 in (8.88) independent?
8 EXERCISES 417
Gina is a golfer who has probability p = .05 on each stroke of making a
“supershot” that gains a stroke over par, probability q = .91 of making
an ordinary shot, and probability T = .04 of making a “subshot” that
costs her a stroke with respect to par. (Non-golfers: At each turn she
advances 2, 1, or 0 steps toward her goal, with probability p, q, or r,
respectively. On a par-m hole, her score is the minimum n such that she
has advanced m or more steps after taking n turns. A low score is better
than a high score.)
a Show that Gina wins a par-4 hole more often than she loses, when
she plays against a player who shoots par. (In other words, the
probability that her score is less than 4 is greater than the probability
that her score is greater than 4.)
b Show that her average score on a par-4 hole is greater than 4. (Therefore
she tends to lose against a “steady” player on total points, although
she would tend to win in match play by holes.)
Exam problems
35 A die has been loaded with the probability distribution
WFJ) = PI ; Pr(m) = ~2; . . . . Pr(m) = p6.
Let S, be the sum of the spots after this die has been rolled n times. Find
a necessary and sufficient condition on the “loading distribution” such
that the two random variables S, mod 2 and S, mod 3 are independent
of each other, for all n.
36 The six faces of a certain die contain the spot patterns
� �☺���☺�
instead of the usual � through � .
a Show that there is a way to assign spots to the six faces of another
die so that, when these two dice are thrown, the sum of spots has the
same probability distribution as the sum of spots on two ordinary
dice. (Assume that all 36 face pairs are equally likely.)
b Generalizing, find all ways to assign spots to the 6n faces of n dice so
that the distribution of spot sums will be the same as the distribution
of spot sums on n ordinary dice. (Each face should receive a positive
integer number of spots.)
37 Let p,, be the probability that exactly n tosses of a fair coin are needed
before heads are seen twice in a row, and let qn = ,&, pk. Find closed
forms for both p,, and qn in terms of Fibonacci numbers.