Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 849 — #857
19.5. Linearity of Expectation 849
Since the groups are chosen randomly, each soldier in the group has the disease
with probability p, and it is safe to assume that whether one soldier has the disease
is independent of whether the others do.
(a) What is the expected number of tests in Approach (2) as a function of the
number of soldiers n, the disease fraction p, and the group size k?
(b) Show how to choose k so that the expected number of tests using Approach (2)
is approximately n p p. Hint: Since p is small, you may assume that .1 p/ k 1
and ln.1 p/ p.
(c) What fraction of the work does Approach (2) expect to save over Approach
(1) in a million-strong army of whom approximately 1% are diseased?
(d) Can you come up with a better scheme by using multiple levels of grouping,
that is, groups of groups?
Problem 19.31.
A wheel-of-fortune has the numbers from 1 to 2n arranged in a circle. The wheel
has a spinner, and a spin randomly determines the two numbers at the opposite ends
of the spinner. How would you arrange the numbers on the wheel to maximize the
expected value of:
(a) the sum of the numbers chosen? What is this maximum?
(b) the product of the numbers chosen? What is this maximum?
Hint: For part (b), verify that the sum of the products of numbers oppposite each
other is maximized when successive integers are on the opposite ends of the spinner,
that is, 1 is opposite 2, 3 is opposite 4, 5 is opposite 6, . . . .
Problem 19.32.
Let R and S be independent random variables, and f and g be any functions such
that domain.f / D codomain.R/ and domain.g/ D codomain.S/. Prove that f .R/
and g.S/ are also independent random variables.
Hint: The event Œf .R/ D a is the disjoint union of all the events ŒR D r for r
such that f .r/ D a.
Problem 19.33.
Peeta bakes between 1 and 2n loaves of bread to sell every day. Each day he rolls