answer the following five problems and show all the work needed.  you
can add comments to explain the logic of the solution.

1) in rolling six true dice, find the probability of obtaining:
a) at least one.
b) exactly one.
c) exactly two, aces.
d) compare your answer with the Poisson approximations.

2) a person purchased 10 of 1000 tickets sold in a certain raffle.  to
determine the five prize winners, 5 tickets are to be drawn at random
without replacement.  compute the probability that the person will win
at least one prize.

3) if the random variable X has a poisson distribution so that
P(X=1)=P(X=2), find P(X=4).

4) the key to a cabinet lock is in a bag with 9 other keys. suppose a
key is randomly selected from the bag and tried in the lock until the
right key is found.
a) assuming that the key is replaced in the bag before the next
selection is made, find the least number N so that the probability of
opening the lock is no more than N trials is at least 75%.
b) do the same if the selected key is discarded rather than being
returned to the bag before the next key is selected.

5) let the probability of exactly one blemish in one foot of wire to
be 1/1000, and let the probability of two or more blemishs in that
length be, for all practical purposes, zero.  assume stochastic
independence of the number of blemishes in nonoverlapping intervals. 
estimate the probability that there are exactly 5 blemishes in 3000 ft
of wire:
a) using poisson distribution.
b) using a binomial approximation.