Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 739 — #747
17.6. References 739
Problems for Section 17.2
Practice Problems
Problem 17.1.
Let B be the number of heads that come up on 2n independent tosses of a fair coin.
(a) PrŒB D n is asymptotically equal to one of the expressions given below.
Explain which one.
1.
2.
3.
4.
1
p
2 n
2
p n
1
p n
q
2
n
Exam Problems
Problem 17.2. (a) What’s the probability that 0 doesn’t appear among k digits
chosen independently and uniformly at random?
(b) A box contains 90 good and 10 defective screws. What’s the probability that
if we pick 10 screws from the box, none will be defective?
(c) First one digit is chosen uniformly at random from f1; 2; 3; 4; 5g and is removed
from the set; then a second digit is chosen uniformly at random from the
remaining digits. What is the probability that an odd digit is picked the second
time?
(d) Suppose that you randomly permute the digits 1; 2; ; n, that is, you select
a permutation uniformly at random. What is the probability the digit k ends up in
the ith position after the permutation?
(e) A fair coin is flipped n times. What’s the probability that all the heads occur
at the end of the sequence? (If no heads occur, then “all the heads are at the end of
the sequence” is vacuously true.)
Class Problems
Problem 17.3.
The New York Yankees and the Boston Red Sox are playing a two-out-of-three