HW 6 solutions

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HW 6 solutions
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1. Voters often change their party affiliation
is subsequent election. In a certain district
republicans remain republicans for the next
election with probability qR 0.9, while
Democrats stay with their party with p
0.8. Describe this process and find the
asymptotic distribution.
Solution Prob(D-gtR) p 0.2 Prob(R-gtD) q
0.1 fD q/(pq) 1/3 fR p/(pq) 2/3.
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2. Four out of every five tracks on the road is
followed by a car, while one out of every six
cars is followed b a truck. What proportion of
vehicles on the road are tracks? t - share of
tucks on the roadc - share of carsp(a,b)-
transition probability for a -gt b (see the table)
t p(c,t) c p(t,t) t (1/6) c (1/5) t
(4.1) t c 1
(4.2) (4.1) can be rewritten
as c/6 4/5 t 0. Using (4.2) we find t
5/29 p/(pq) Note You can also solve it with
Mathematica using the Solve function(try it!)
c t
c 1-p5/6 p 1/6
t q 4/5 1-q 1/5
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3. You are about to have an interview for
Harvard Law school. 60 of the interviewers are
liberals and 40 are conservatives. 50 of
conservatives smoke cigars but only 25 liberals
do. The interview lights up a cigar. What is the
probability that he is liberal? P(LS) P(L
S)/(P(L S) P(C S)) 15/(15 20) 3/7.
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4. Suppose for simplicity that number of children
in a family is 1 , 2 or 3 with probability 1/3
each. Little Bobby has no sisters. What is the
probability that he is the only child? 1B Bob
the only child (and he is a boy!) NS No
sisters. In other words, we need the probability
of 1 boy given that there could be 1 or 2 or 3
boys. P(1BNS) P(1B NS)/P(NS)
P(1B)/P(NS).P(NS) P(1B) P(2B) P(3B) 1/3
(1/2 1/4 1/8) 7/24P(1B) 1/31/2
1/6. P(1BNS) (1/6)/ (7/24) 4/7
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5. Plumber Bob does 40 of the plumbing jobs in a
town. 30 of people in town are unhappy with
their plumbers but 50 of Bobs customers are
unhappy with his work. If your neighbor is
unhappy with his plumber, what is the probability
it was Bob? P(BU) P(B U) /P(U) 0.4 0.5/0.3
2/3