Math 1180 Lab 08 Homework Problems
Lab meets: March 9, 2011
Report due date: March 16, 2011
Instructions: Complete each of the following exercises. Use Maple as needed. In your write-up, put your
Maple output/graphs with (or very near) their corresponding problems. Attach any Maple code you write AT
THE END of your assignment.
1. After a day of growth, the measurement L describing the length of a group of worms has p.d.f
f (l) = 1.5(l + l2 )
for 0.5 ≤ l ≤ 1.0.
(a) Graph the p.d.f. and check that it is consistent.
(b) Find the probability that a worm is less than 0.75 mm long.
(c) Find the approximate probability that a worm is between 0.75 and 0.76 mm long. Sketch the
region corresponding to this probability on a graph of the p.d.f.
(d) How many worms out of 1000 would you expect to be longer than 0.75 mm?
2. During a 10-min interval, adult worms switch from eating to egg-laying with probability 0.1 and from
egg-laying to eating with probability 0.15.
(a) Draw a diagram illustrating this process.
(b) Derive a discrete-time dynamical system for the probability that the worm is eating.
(c) Find the long-term fraction of time spent eating.
(d) If egg-laying takes place at a rate of 1.5-eggs per 10 min, about how many eggs would this worm
lay during its 15-day life? (Assume worms never sleep.)
(e) Simulate this experiment for a day, i.e. draw random numbers in Maple to decide what the worm
was doing in each 10-min period over 24 hours. Show the number of intervals the worm spent
eating and egg-laying in a histogram. What fraction of time did the worm spend eating in your
experiment?
3. Two codons of DNA are being compared (each consists of three nucleotides). The experiment is to
check whether the two match. For example, if the first is AAG and the second ATG, the result would
be “match, no match, match.”
(a) List all of the simple events.
(b) What simple events are included in the event “they match at the second nucleotide”?
(c) Find the union and intersection of the events “they match at the second nucleotide” and “they
do not match at the first nucleotide.”
(d) Suppose the probability of a change is 0.1 at the first nucleotide, 0.2 at the second, and 0.4 at the
third. What is the probability of no change at any site if the events are independent?
4. A laboratory is testing for a rare bacterial mutant, which appears in 1% of cells. A test correctly
identifies a mutant with probability 0.9 and gives a false positive with probability 0.02.
(a) Find the probability of a positive test.
(b) Find the probability that a cell that tests positive has the mutation.