Suppose you are on a game show, and you are given the choice of three doors. You win
what is behind the door you choose. Behind one door is a new car; behind the other two doors
are goats. You pick a door, say door number 1. Afterwards, the game show host, who knows
what is behind all the doors, opens another door, say door number 3, and shows you a goat.
He says to you, “Do you want to change your pick to door number 2?” Is it to your advantage to switch your choice of doors? Why?

Three out of every four trucks on the road are followed by a car, while only one out of
every five cars is followed by a truck. What proportion of vehicles on the road are trucks?

A knight is placed alone on one of the corner squares of a chessboard (having 8 x 8 = 64
squares). What is the expected total number of moves for the knight to return to its initial
position, if we assume that the knight moves randomly, taking each of its legal moves in each
step with equal probability?

A truck driver regularly drives round trips between Atlanta (A) and Boston (B). Each
time he drives from A to B, he drives at a fixed speed that (in miles per hour) is uniformly
distributed between 40 and 60. Each time he drives from B to A, he drives at a fixed speed
that (in miles per hour) is equally likely to be either 40 or 60.

In the long-run, what proportion of his driving time is spent going from A to B?

Consider successive independent flips of a biased coin. On each flip, the coin comes up
heads (H) with probability p or tails (T) with probability q = 1 ¡ p, where 0 < p < 1. A
given segment of ¯nitely many consecutive outcomes is called a pattern. The pattern is said to
occur at flip n if the pattern is completed at flip n. For example, the pattern A = HTHTHT
occurs at flips 8 and 10 in the sequence TTHTHTHTHTTTTHHHT … and at no other
times among the first 17 flips.

What is the expected number of flips until the pattern HHH first appears?

Let T_a be the first time standard Brownian motion hits the state a. Compute

The Columbia hockey team is playing a game against Cornell: it will either win, lose or
draw. A gambler offers you the following three payoffs, each for a $1:00 bet.
win lose draw
bet1 0 1 1:5
bet2 2 2 0:0
bet3 0:5 1:5 1:0
Assume that you are able to buy any amounts of these bets (even negative). Is there an
arbitrage opportunity?