N balls labeled 1 through are placed in Box 1 while a Box 2 is initially empty.

At each time instant, one of the balls is chosen (with equally probability) and moved to the other box.

  be the number of balls in Box 1 at time instant .

Draw a state diagram and find the transition probability matrix for this Markov chain.

Note: This is known as the Ehrenfest chain and was developed by the dutch Physicist Paul Ehrenfest for the study of molecular dynamics.