Question

Professor Hay never orders eggs two days in a row, but if he orders tofu one day, then there is an equal probability that he will order tofu or eggs the next day.

Find the following:

1. If Professor Hay had eggs on Monday, what is the probability that he will have tofu on Friday?

2. Find the long term distribution for breakfast choices for Professor Hay.

Answer 1

here transition mtrix T =
	eggs	tofu
eggs	0	1
tofu	0.5	0.5

a)

for monday initial state vector Xo=[ 1 0]

therefore state vector on Friday (after 4 days) =X0*T⁴ =[3/8 5/8]

therefore probability that he will have tofu on Friday =5/8 =0.6250

b)

let fixed probability vector is W = [a 1-a]
therefore from relation: WT =W
0a+0.5(1-a)=a …….(1)
solving above: a=1/3

long term distribution for breakfast choices for Professor Hay =[1/3 2/3}

long term probability of eggs =1/3

long term probability of tofu =2/3