Here is a simulation of the Enrenfest's Urn stochastic model. (If you have trouble running the applet, see these notes.)

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At each iteration, one of the balls is chosen uniformly at random, and swapped to the opposite urn.

The "state" is the number of balls currently in Urn #1.

Does the state converge in probability to some distribution?

The applet accepts the following keyboard inputs. (You may need to "click" on the applet first.)

Applet by Jeffrey S. Rosenthal (contact me). See also my Applets Page / Home Page