Assignment 1
(1) In an attempt to save money on screening people for AIDS through
expensive blood tests, the government has come up with the following
idea. Instead of testing the blood of each individual separately, the
blood of a group of people are combined and the mixture tested. If the
test results are negative, all those tested are free of infection, and
a lot of money is saved by not testing individually. However, if the
test comes out positive, everyone in the group has to be tested. Under
what conditions will this testing strategy save money, i.e., when will
the expected number of tests using groups be less than the number of
tests required without grouping.
(2) A soft drink company is capping their bottles with 10 different
types of crowns, each type of crown being as likely to be used as any
other. How many bottles will you have to drink to get at least one
crown of each type. Write a program (in Matlab or any other language
you know) to verify your answer. Submit the program as well as the
output with your assignment.
(3) You are measuring the height of a group of children - their heights
are normally distributed with a mean of 1.3 meters and a standard
deviation of 0.5 meters. If you pick a child at random from this
population, what is the probability that its height is between 1.6 and
1.9 meters ? What is the probability that its height is exactly 0.9 meters ?
(4) Suppose that 10^10 (10 raised to 10) monkeys have been seated at
typewriters throughout the age of the universe, 10^18 s. Suppose a
monkey can hit 10 typewriter keys per second (A typewriter is assumed
to have 44 keys). If the Mahabharata
has 10^5 characters, will the monkeys accidentally compose the Mahabharata ?
Show that the probability that any given sequence of 10^5 characters
typed at random will come out in the correct sequence (of the Mahabharata) is of the order of
10^-164345 [using the result that log_10 (44) = 1.64345].
(5) Find the entropy of a set of N [quantum] harmonic oscillators of
frequency w as a function of the total quantum number n. Use the
multiplicity of states expression derived in class (3/2/06) and make
the Stirling approxn log N! = NlogN - N (you can replace N-1 by N).
MATLAB is available in imsc7
and imsc8