Interesting problems in
probability
Important:
while the answers to the problems below may be found in books, you are
encouraged to solve them yourselves!
- Umbrella problem
I do not like carrying an umbrella with me. After all, it does not rain here that often. And yet I do not like to get wet. This morning, when I woke up, I had x>0
umbrellas at
home
and y>0
umbrellas in my
office.
Every day, before I leave for work, I look out of the window to see if
it is
raining, and if it is, I take one of the umbrellas with me to work,
where I would leave it. Similarly, when I go home,
I look out of the window and if it rains, I take
one of the umbrellas with me home. Thus the number of the
umbrellas staying at home or in my office fluctuates, but totally I
still have x+y umbrellas. Of
course, one
day,
it will turn out that I have to go outside in the rain, but all these
umbrellas are not where I am at the moment. Assuming that the
probability of a rain before each of my departures is p>0
and these
event are independent, find the expected number of days before this
unfortunate event (=I have to go under the rain without an umbrella)
happens. Also find the probability that I am at home and not in my
office at that moment
- Banach
match box problem
Professor Banach still has a bad habit of smoking. So, he carries two
matchboxes, one in his left pocket, and one in the right one, each
initially having N>0
matches. Whenever he wants to light a
cigarette, he takes one match from a matchbox, and this matchbox is
equally likely to be in either of the pockets. One day, it turns out
that the matchbox he chose is already empty. Find the
distribution function of the number of matches remaining in the other
match box. (Hint: this will be a random variable X, taking
values 0,1,2,...,N).
- Sums
of
uniform random variables
Suppose
U1, U2, U3, ... are independent
identically distributed,
uniform [0,1] random
variables. How many of them do you have to sum up on average, before
there sum exceeds a=1?
Formally, let
N=min{k: U1+U2+...+Uk>1}.
Find E(N).
What can you say if I replace a=1
by some other positive number?
Warning: the sum of uniform random variables is not a uniform
random
variable!
- Birthday
problem
You already know that if there are N=23
people in your class, the
probability that at least two of them have their birthdays on the same
day, is close to ½.
What is the probability that at least three of them
have the birthdays on the same day? How large N should be so
that this
probability is also close to ½?