## Instructions

Homework 3 can be downloaded here on the ECE 302 course website.

## Problem 1: Monte Hall, twisted

http://nostalgia.wikipedia.org/wiki/Monty_Hall_problem Explains the original Monty Hall problem and then the problem considering two contestants are involved.

Consider the following twist on the monty hall problem (see video above to recollect):

There are 3 doors, behind one of which there is a car, and the remaining two have goats. You and a friend each pick a door. One of you will have definitely picked a goat. The host then opens one of the two chosen doors (i.e. yours or your friend's), shows a goat and kicks that person out (if both of you have chosen goats, he opens one door at random). The other person is given the option of staying put, or switching to the one remaining door.

• (a) When originally picking your doors, should you choose your door first, or let your friend go first? Does it make a difference?
• (b) It turns out that the host has just eliminated your friend. What should you do? (i.e. stick with your original door, or switch?) What are the probabilities of winning in each case?

## Problem 2: A Bayesian Proof

The theorem of total probability states that $P(A) = P(A|C)P(C) + P(A|C^c)P(C^c)$. Show that this result still holds when everything is conditioned on event $B$, that is, prove that

$P(A|B) = P(A|B\cap C)P(C|B) + P(A|B\cap C^c)P(C^c|B).$

## Problem 3: Internet Outage

A certain Internet service provider in a midsize city relies on $k$ separate connections between the city and neighboring cities, to connect its customers to the Internet. Based on past experience, management assumes that a given connection will be down on a given day with probability $p = 0.001$, independently of what happens on other days or with other connections. Total outage is said to occur if all connections are down on the same day. How large should $k$ be so that the probability total outage occurs at least one day in a year is less than or equal to 0.001?

## Problem 4: Colored Die

There are three dice in a bag. One has one red face, another has two red faces, and the third has three red faces. One of the dice is drawn at random from the bag, each die having an equal chance of being drawn. The selected die is repeatedly rolled.

• (a) What is the probability that red comes up on the first roll?
• (b) Given that red comes up on the first roll, what is the conditional probability that red comes up on the second roll?
• (c) Given that red comes up on the first three rolls, what is the conditional probability that the selected die has red on three faces?

## Problem 5: Fuzzy Logic

As we saw in class, two random bits are independent if knowing the value of one does not change your beliefs about the other. We saw that if $A$ and $B$ are independent and each is equally likely to be 0 or 1, then $A$ XOR $B$ is independent of $A$. Is the same true if the bits are biased, with the probability of being 1 equal to $p$?

## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett