Homework 1 (Due Sunday April 3 at midnight)

The exercise this first week is to help you develop your intuition in
probability for games of chance.  In class, we played a game which is
similar to the "Monty Hall Problem", described as follows
(from "Power of Logical Thinking" by Marilyn vos Savant):

"Suppose you're on a game show, and you're given the choice of three doors.
Behind one door is a car, behind the others, goats. You pick a door, say
number 1, and the host, who knows what's behind the doors, opens another
door, say number 3, which has a goat. He says to you, "Do you want to pick
door number 2?" Is it to your advantage to switch your choice of doors?"

1. What do you think your chances are of getting the car if you do not switch?
What about if you do switch?

2. Test your intuition by actually playing the game with a friend.  If you
cannot find someone to play with, go the the following web site which
simulates the game: http://math.ucsd.edu/~crypto/Monty/monty.html   You
may not believe the simulation, so playing with a friend is best
(and more fun).
Play the game at least 60 times, where 30 times you do not switch, and
the other 30 times where you do switch.

What was the "win rate" (games won divided by total games played) when you
used the no-switch strategy?

What was the win rate when you used the switch strategy?

3. Explain why the win rates come out the way they do?

Try to do this exercise without doing any further research (on the web, etc.)
on this problem.  I want you to take a lot of time thinking about this
problem and trying to figure out why the win rates work out the way they do.
Afterwards, I will give you pointers to solutions to this problem, but it is
important for you to struggle through this on your own before reading what
others say.

Send me your write-ups via email (pasquale at ucsd.edu).  I'm much more
interested in how you came up with your solution and how you developed
your intuition, and what aspects of the problem you most struggled with,
rather than simply producing "the right answer."