MATH 3900, Spring 2009
Homework 9
Due 04/03/09
1. A box has 4 balls: 2 blue and 2 white. You pick two at random. What is
the probability that both will be blue? Solve this in two ways:
a) Compute probability analytically, similar to what we did in class using
“n choose k”. Compare the result to what we found in class making a list of all
possible outcomes.
b) (computing) Do a ”’Monte-Carlo”’ simulation of this experiment, namely,
pretend the balls are numbered from 1 to 4 (with blues being, say 1 and 2), and
ask the Matlab to draw 2 of them at random, say, 10 times, and each time
record whether you got the desired result (blue-blue) or not. From this, make
an estimate of the probability that both drawn balls are blue. Do this with 10,
100 and 1000 trials and compare results with your theoretical result from a).
There is a code on the homework page that you can use as a part of your code.
2. Consider the Awake-Sleep Markov chain model that we discussed in
class. Consider the following transition probabilities: PAA = 0.9, PAS = 0.1,
PSA = 0.4,PSS = 0.6.
a) Starting with p0A = 1, and p0S = 0, find the fraction of awake and sleeping
students at several subsequent time steps. Repeat iteration until the distribution
stabilizes. What are the resulting awake and asleep fractions? Does it match
the theoretical steady state we found in class?
b) (computing) Do a Monte-Carlo (random) simulation of one student.
Start in awake state. Then ask matlab to toss the appropriate coin and decide
which state the student will be in at the next time step. Record the sequence
of states for 20 time steps. What proportion of time was the student awake?
Compare it with the steady state you estimated from a).
3. The model for random genetic drift in a population of bacteria that I
described in class has the following ingredients: The states of the markov chain
are marked by the number of A-type plasmids in the cell. There are N+1 states
(the number of A plasmids can vary from 0 to N). In the n-th generation a
fraction of cells in each state is p[n] = (pn0 , . . . , pnN ). In the next generation
p[n+1] = p[n] P
where P is the transition matrix given by
2i 2N −2i
Pi,j =
j
1
N −j
2N
N
,
where
n
0
= 1 and
n
k
= 0 for k > n, and i and j vary from 0 to N .
a) Consider the case when each cell has 4 plasmids. In the original cells
we introduce half of the plasmids of type A and half of the plasmid of type
B, so p02 = 1. Iterate to find what will be the distribution of plasmids A and
B in the population after 50 generations. This phenomenon was first observed
experimentally and it was hypothesized that the plasmids interact so that they
cannot be picked up into the same cell. Your result shows that the separation
of plasmid types can happen simply as a result of random plasmid transfer.
b) Based on your previous result, if plasmid type A carries resistance to an
antibiotic gene, and plasmid type B does not, and you need a population of
10,000 antibiotic resistant cells, how many cells total do you need to grow?
(extra credit) c) Do a monte-Carlo simulation of 100 cells, starting them
with 2 plasmids of type A and 2 of type B and watch how the distribution of
plasmid types evolves in the population. Come up with a way to visulaize it in
ine picture.
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