1. Consider the following randomized algorithm for generating biased random bits. The subroutine FAIRCOIN returns either 0 or 1 with equal probability; the random bits returned by FAIRCOIN are mutually independent.
ONEINTHREE:
if FAIRCOIN = 0
return 0
else
return 1 - ONEINTHREE
(a) Prove that ONEINTHREE returns 1 with probability 1/3.
(b) What is the exact expected number of times that this algorithm calls FAIRCOIN?
(c) Now suppose you are given a subroutine ONEINTHREE that generates a random bit that is equal to 1 with probability 1/3. Describe a FAIRCOIN algorithm that returns either 0 or 1 with equal probability, using ONEINTHREE as your only source of randomness.
(d) What is the exact expected number of times that your FAIRCOIN algorithm calls ONEINTHREE?

2.Consider the following algorithm for finding the smallest element in an unsorted array:
RANDOMMIN(A[1 .. n]):
min ← ∞
for i ← 1 to n in random order
if A[i] < min
min ← A[i] ( )
return min
(a) In the worst case, how many times does RANDOMMIN execute line ( )?
(b) What is the probability that line ( ) is executed during the nth iteration of the for loop?
(c) What is the exact expected number of executions of line ( )