Question

Nitially, Anthony has n camels, and Cleopatra has n horses. Every day, each of them selects one of their animals at random, and sends it to the other as a gift. Let Xm denote the number of camels that Cleopatra has urn after m days.

Find the transition matrix for this chain.

Answer 1

The transition matrix for this chain can be represented as A = [p00 p01; p10 p11], where p00 represents the probability that Anthony has n camels and Cleopatra has n horses after m days.

Step-by-step explanation:

The transition matrix for this chain can be represented as:

A = [p00 p01; p10 p11]

where p00 represents the probability that Anthony has n camels and Cleopatra has n horses after m days, p01 represents the probability that Anthony has n camels and Cleopatra has n+1 horses after m days, p10 represents the probability that Anthony has n+1 camels and Cleopatra has n horses after m days, and p11 represents the probability that Anthony has n+1 camels and Cleopatra has n+1 horses after m days.

The value of p00 is (n-1)/(2n-1), the value of p01 is 1/(2n-1), the value of p10 is 1/(2n-1), and the value of p11 is (n)/(2n-1).