1.
Suppose you are playing on a one-armed bandit. Each game costs $ {\$}1$, and you have one chance in a million of winning $ {\$}999,999$. What is your expected loss after one million games. Find the distribution of your loss after one million games. What is your chance of losing exactly $ {\$}1$ in a million games. What is your chance of losing exactly $ {\$}1000,000$. Use Chebyshev's inequality to give an estimate of how many games you must play until you have a $ 90{\%}$ or better chance that your loss or profit in dollars is less than a quarter of the number of games played.

2.
Consider the following game, where your winnings depend on the number of games already played. On the $ n^{\text{th}}$ game you win $ n$ dollars with probability $ \frac{1}{2 n^2}$, you lose $ n$ dollars with probability $ \frac{1}{2 n^2}$, or else you win nothing. Let $ S_n$ denote your total winnings after $ n$ games, and let $ \mu_n$ denote your expected average winnings after $ n$ games. Show that for any $ \epsilon > 0$,

$\displaystyle \Prob\left(\left\vert \frac{S_n}{n} - \mu_n \right\vert \geq \epsilon\right) \rightarrow
0.
$

In other words, show that your winnings satisfy the weak law of large numbers.

3.
Introduction to Probability, Section 8.1, Exercises 4, 8.

4.
Introduction to Probability, Section 9.1, Exercises 1,2,6,14,17.

5.
Ceiddwen starts her car, puts her `The Velvet Underground' CD into her car CD player, and pushes the continuous random play button, which repeatedly chooses a track to play at random (choosing from all tracks on the disc each time). She then heads down to Washington DC. See the accompanying CD cover for a list of tracks. Elcin asks her if he can drive instead, so as a compromise she suggests: `Let's switch after 60 tracks. OK?'. There is a two second gap between each track, and it took two seconds for the CD to start.

a. What is the probability that Elcin has to wait longer than four and three quarter hours, before he's allowed to drive? What is the probability that he has to wait less than four and a quarter hours? What is the probability that the switch happens between four hours twenty minutes and four hours forty minutes?

b. Suppose you discover that Ceiddwen was driving for at least the first four and a half hours of the trip (e.g. you couldn't reach her on her cell phone). What is the probability that `That's the story of my life' was never played? What is the probability that `That's the story of my life' was played exactly ten times?

When she arrives in DC, Ceiddwen decides to test her friend Neepa's problem solving skills. She tells her that 60 tracks were played, that the switch took place more than four and a half hours after the start of the trip, and that the CD played was either The Velvet Underground or Infected. Neepa works out the probability of each CD using Bayesian analysis.

c. What probabilities does Neepa calculate? Is Ceiddwen's question fair?



Math 20 Fall 1998
David K. Maslen
1998-11-08