CS174 Midterm Sept. 25, 2003
1. Let A and B be events in the same sample space, such that A2/5, B3/10
and (A|B) = 2/3. What is (B|A)?
2. In a permutation = ((1), (2), ..., ()), index i is called a cumulative maximum if
(i)=max((1),(2),...,()). What is the expected number of cumulative maxima in a
random permutation of {1,2,...,}?
3. An airline knows that on average five percent of the people making reservations on a
certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight
that can only hold 50 passengers. Give a formula involving binomial coefficients (but not
involving the summation symbol) for the probability that there will be a seat available for
every passenger that shows up.
4. Let be a set of elements. At the first stage each element in is independently
removed with probability . Those elements not removed constitute the set . If is not
the empty set then each of its elements is independently removed with probability , with
the remaining elements constituting the set , and so on. Let be the least such that
is empty. Give a formula for the probability that = . Hint: can be expressed as the
maximum of a set of independent geometric random variables.
5. A rat is trapped in a maze. Initially it has to choose one of two directions. If it goes to
the right it will wander around in the maze for three minutes and will then return to its
initial position. If it goes to the left then with probability 1/3 it will depart the maze after
two minutes of traveling, and with probability 2/3 it will return to its initial position after
five minutes of traveling. Assuming that the rat is at all times equally likely to go to the
left or to the right, give the expectation of the number of minutes that it will be trapped in
the maze.
6. Suppose that is a random variable with mean 10 and variance 15. What can we say
about (5<=<=15)?