Spring 2016 Math 166
3. Market studies indicate the following
trends in car insurance:
Of those who use Farmers' Insurance,
30% switch to GEICO the next year
and 20% switch to Progressive.
Of those who use GEICO, 5% switch to
Farmers the next year and 10% switch
to Progressive
Of those who use Progressive, 5%
switch to Farmers the next year and
25% switch to GEICO.
Week in Review XIII
courtesy: David J. Manuel
(covering Sections M.1-M.3)
1
Section M.1
1. The tree diagram below represents
the initial state of a given event A
and the subsequent probabilities of
changing to/from event A periodically:
0.8
(a) If 50% are insured by Farmers, 30% by
GEICO, and 20% by Progressive, what
percentage will be insured by GEICO
the next year?
A
A
0.2
0.1
0.9
0.6
c
(b) What percentage will be insured by
Farmers in 2 years?
A
(c) What will the percentage breakdown be
in 5 years?
A
c
A
c
0.4
A
4. The College of Engineering at A&M
(a) Assuming the subsequent probabalities
continue, write the transition matrix T
for this process.
has a Lower Division (those who have
not completed their CBKs) and an
Upper Division (those who have completed their CBKs and focus on their
major). Among those in the Lower Division, 30% remain in the Lower Division after one year, 40% drop out of
Engineering, and 30% move to Upper
Division. Among those in the Upper
Division, 10% drop out of Engineering,
70% remain in the Upper Division, and
20% Graduate.
(b) Find the probability of A after the fth
process.
2. An insurance company found that,
over a period of time, 29% of the
drivers in a town who were involved
in an accident one year were involved
in an accident the following year. They
also found that only 10% of the drivers
who were not involved in an accident
one year were involved in an accident
the following year.
(a) Write the transition matrix for a student's status in the College of Engineering.
(a) If 5% of the town are involved in an accident this year, what percentage are expected to be involved in an accident the
following year?
(b) What is the probability that a student
entering the College will graduate in 4
years?
(b) What percentage are expected to be involved in an accident in 3 years?
1
2
Find the limiting matrix and interpret
each entry in the upper right block.
Section M.2-M.3
1. Determine which of the above pro-
cesses are regular Markov processes.
For those that are, give the steadystate. For those that aren't, nd and
interpret the limiting matrix.
(a)
(b)
(c)
(d)
#1
#2
#3
#4
2. An auto insurance company classies
its customers in one of three categories:
poor drivers, satisfactory drivers, and
preferred drivers. Each year, 30% of
poor drivers move to satisfactory, and
15% of satisfactory drivers move to
preferred. In addition, 20% of preferred drivers are pushed back to satisfactory, and 10% of satisfactory drivers
are pushed back to poor. Customers
are never moved between poor and preferred in one year.
(a) Assuming all customers start as satisfactory drivers, what percentage of new
customers in a given year will be in each
category after 3 years?
(b) Show that this is a regular Markov process.
(c) Find the steady state.
3. A credit union classies auto loans into
one of four categories: (F)=paid in full,
(G)=good standing, (A)=in arrears
(missing payments), or (B)=bad debt
(turned over to a collection agency).
Past history indicates the following
changes:
(G): 10% become paid in full, 20% become in arrears
(A): 5% become paid in full, 40% become in good standing, 20% become
bad debts.
2