#### CSCI 6454 - Advanced Algorithms - Spring 2014

### Homework #1

#### Due: Feb 10th, 2014

1. I have 9 fair coins and a 10th coin where both sides are heads. I select a coin
at random and flip it; it comes out as heads. What is the probability this is the
coin with both sides heads?
2. Give an example of three random events $X$, $Y$, and $Z$ where every pair is
independent but all three are not mutually independent.

3. Suppose you and a friend are shipwrecked on a desert island and you want to flip
a coin to decide who is going to swim to a nearby island for help. Unfortunately
you don't have a coin to flip, but there are some flat rocks with distinctive markings.
However, these rocks are probably biased.

Given an algorithm that allows you to use a flat rock to generate fair coin flips.
Implement your algorithm and run 1,000,000 trials to show that it's (very close to) 50/50.
Turn in your algorithm and its output.

4. A medical company touts its new test for a certain genetic disorder. The false
negative rate is small: if you have the disorder, the probability that the test
returns a positive result is 0.999. The false positive rate is also small: if you do not
have the disorder, the probability that the test returns a positive result is only
0.005. Assume that 2% of the population has the disorder. If a person chosen uniformly
from the population is tested and the result comes back positive, what is the probability
that the person has the disorder?

5. Implement the randomized min-cut algorithm in a language of your choice and run it
on this graph. Run 1000 iterations and output
the size of the min cut you find. Turn in your code and the output. You are free to
compare your answer to others in the class. (Note: the file may contain parallel edges, but
this is fine since your data structure will need to accommodate parallel edges anyway.)