Some of the best probability problems I have ever seen

Some awesome probability problems from ucdavis-
1. [P. Winkler] One hundred people line up to board an airplane. Each has a boarding pass with
assigned seat. However, the first person to board has lost his boarding pass and takes a random
seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied
seats at random otherwise. What is the probability that the last person to board gets to sit in
his assigned seat?
2. [D. Knuth] Mr. Smith works on the 13th floor of a 15 floor building. The only elevator
moves continuously through floors 1
, 2, . . . , 15, 14, . . . , 2, 1, 2, . . . , except that it stops on a floor
on which the button has been pressed. Assume that time spent loading and unloading passengers
is very small compared to the travelling time.
Mr. Smith complains that at 5pm, when he wants to go home, the elevator almost always
goes up when it stops on his floor. What is the explanation?
Now assume that the building has
n elevators, which move independently. Compute the
proportion of time the first elevator on Mr. Smith’s floor moves up.
3. [D. Barsky]
NCAA basketball pool . There are 64 teams who play single elimination tourna-
ment, hence 6 rounds, and you have to predict all the winners in all 63 games. Your score is
then computed as follows: 32 points for correctly predicting the final winner, 16 points for each
correct finalist, and so on, down to 1 point for every correctly predicted winner for the first
round. (The maximum number of points you can get is thus 192.) Knowing nothing about any
team, you flip fair coins to decide every one of your 63 bets. Compute the expected number of
points.
4. [E. Berlekamp]
Betting on the World Series. You are a broker; your job is to accommodate
your client’s wishes without placing any of your personal capital at risk. Your client wishes to
place an even $1,000 bet on the outcome of the World Series, which is a baseball contest decided
in favor of whichever of two teams first wins 4 games. That is, the client deposits his $1,000
with you in advance of the series. At the end of the series he must receive from you either $2,000
if his team wins, or nothing if his team loses. No market exists for bets on the entire world series. However, you can place even bets, in any amount, on each game individually. What is your strategy for placing bets on the individual games in order to achieve the cumulative result demanded by your client?
5. From
New York Times, Science Times, D5, April 10, 2001:
“Three players enter a room and a red or blue hat is placed on each person’s head.
The color of each hat is determined by [an independent] coin toss. No communication of any sort is allowed, except for an initial strategy session before the game begins.
Once they have had a chance to look at the other hats [but not their own], the
players must simultaneously guess the color of their own hats or pass. The puzzle is to find a group strategy that maximizes the probability that at least one person guesses correctly and no-one guesses incorrectly.”
The naive strategy would be for the group to agree that one person should guess and the others pass. This would have probability 1/2 of success. Find a strategy with a greater chance for success.
For a different problem, allow every one of
n people to place an even bet on the color of his
hat. The bet can either be on red or on blue and the amount of each bet is arbitrary. The group wins if their combined wins are strictly greater than their losses. Find, with proof, a strategy with maximal winning probability.
6. [L. Snell] Somebody chooses two nonnegative integers
X and Y and secretly writes them on
two sheets of paper. The distrubution of (
X, Y ) is unknown to you, but you do know that X
and Y are different with probability 1. You choose one of the sheets at random, and observe
the number on it. Call this random number
W and the other number, still unknown to you, Z.
Your task is to guess whether
W is bigger than Z or not. You have access to a random number
generator, i.e., you can generate independent uniform (on [0
1]) random variables at will, so
your strategy could be random.,
Exhibit a stategy for which the probability of being correct is 1
/2 + , for some > 0. This may depend on the distribution of (X, Y ), but your strategy of course can not.

Some of the best probability problems I have ever seen

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