Question

y Notes Ask Your Teacher In a population of 10,000, there are 5000 nonsmokers, 2500 smokers of one pack or less per day, and 2500 smokers of more than one pack per day. During any month, there is a 5% probability that a nonsmoker will begin smoking a pack or less per day, and a 3% probability that a nonsmoker will begin smoking more than a pack day. For smokers who smoke a pack or less per day, there is a 10% probability of quitting and a 10% probability of increasing to more than a pack per day. For smokers who smoke more than a pack per day, there is a 9% probability of quitting and a 10% probability of dropping to a pack or less per day How many peope i į be in each group in 1 month, in 2 months, and in 1 year? (Round your answers to the nearest whole number) (a) in 1 month people people people 1 pack/day or less more than 1 pack/day (b) in 2 months nonsmokers people people people 1 pack/day or less more than 1 pack/day (c) in 1 year 1 pack/day or less more than 1 pack/day people people people

Answer 1

let nonsmokers ; smoking less than 1 pkt and smokeing more than 1 pkt are N, S and H

hence below is transition rmatrix :T

	N	S	H
N	0.92	0.05	0.03
S	0.1	0.8	0.1
H	0.09	0.1	0.81

initial vector X=[ 0.5 25 0.25 ] (as proportion of N,S and H are 0.5,0.25 ,0.25.

hence after 1 month distribution will become

X1 =xT =[0.5075 0.25 0.2425]

non smokers			5075	people
1 pac ket o less			2500	people
more than 1 pkt			2425	people

b)

for two months

X2 =XT² = [0.513725 0.249625 0.23665 }

non smokers			5137	people
1 pac ket o less			2496	people
more than 1 pkt			2367	people

c)

in 1 year: X12 =XT¹² =[0.53921 0.24435 0.21644]

non smokers			5392	people
1 pac ket o less			2444	people
more than 1 pkt			2164	people