- 1.
- Suppose you are playing on a one-armed bandit. Each game costs ,
and you have one chance in a million of winning
. 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
in a million games. What is your chance of losing exactly
.
Use Chebyshev's inequality to give an estimate of how many games you must
play until you have a
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
game you win
dollars with
probability
, you lose
dollars with probability
, or else you win nothing. Let
denote your total
winnings after
games, and let
denote your expected average
winnings after
games. Show that for any
,
- 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?