Mathematics for Computer Science

18.8. Really Great Expectations 649

(b)What value ofb 0 in part (a) gives the best bound?

Problems for Section 18.4

Practice Problems

Problem 18.3.
A gambler plays 120 hands of draw poker, 60 hands of black jack, and 20 hands of
stud poker per day. He wins a hand of draw poker with probability 1/6, a hand of
black jack with probability 1/2, and a hand of stud poker with probability 1/5.

(a)What is the expected number of hands the gambler wins in a day?

(b)What would the Markov bound be on the probability that the gambler will win
at least 108 hands on a given day?

(c)Assume the outcomes of the card games are pairwise independent. What is
the variance in the number of hands won per day?

(d)What would the Chebyshev bound be on the probability that the gambler will
win at least 108 hands on a given day? You may answer with a numerical expres-
sion that is not completely evaluated.

Problem 18.4. (a)A computer program crashes at the end of each hour of use with
probability1=p, if it has not crashed already. IfHis the number of hours until the
first crash, we know

ExŒHçD

1

p

; (Equation (17.8))

VarŒHçD

q

p^2

(Equation (18.8));

whereqWWD 1 p.

(b)What is the Chebyshev bound on

PrŒjH.1=p/j> x=pç

wherex > 0?

(c)Conclude from part (b) that fora 2 ,

PrŒH > a=pç

1 p

.a1/^2

Hint:Check thatjH.1=p/j> .a1/=piffH > a=p.