Transition matrix_jogging bare foot_two doors IAI'13_Q4

[deepakraomore]

Each morning Samit leaves his house and goes for a jog. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of sports shoes (or goes for a jog barefoot if there are no shoes at the door from which he left). On his return he is equally likely to enter, and leave his sports shoes, either by the front or back door.
i) Write down the transition matrix of the process Xn, where Xn represents the number of shoes at the front door.

ii) If he owns a total of pairs of sports shoes, what proportion of the time does he jog barefooted?

Sol-
Let \(X_n\) be the number of shoes at the front door. Then \(X_n\in \{ 0, 1, 2, ...\}\) a Markov chain.
The probability of the Samit jogging without shoes would be equal to:
50% x Probability that there are no shoes at the front door when he leaves (in a steady state)
+
50% x Probability that there are no shoes at the back door when he leaves (in a steady state)
And transition probabilities are \(P_{00} = 0.75\), \(P_{01} = 0.25\), \(P_{10} = 0.25\) \(P_{11} = 0.5\) etc...
how these probabilities are calculated?

 

[divyam sankharva]

\(P_{00}\\=P(\text {When samit returns home after jog there are no pair of shoes at front door given that there were no pair of shoes when he left for jogging})\\=P(\text{He leaves his house from back door and returns from back door})\\+P(\text{He leaves his house from front door and returns from front door})\\+P(\text{He leaves his house from front door and returns from bank door})\\=P(\text{He leaves his house from back door})\times P(\text{He returns from back door})\\+P(\text{He leaves his house from front door})\times P(\text{He returns from front door})\\+P(\text{He leaves his house from front door})\times P(\text{He returns from bank door})\\=0.5\times0.5+0.5\times0.5+0.5\times0.5\\=0.75\)

 

[divyam sankharva]

Similarly we can find the remaining probabilities.
Even though here Xn is defined as no of shoes, the solution given is for number of pair of shoes.