# Conclusions (for both models)

• The transition probability matrix is easier to construct by thinking of it as a product of simpler matrices.

• Knowledge of the transition probability matrix can help determine useful and interesting information about the Markov chain.

• Convergence to the limiting distribution is "good at about 24 turns [about 29 rolls]" (Ash and Bishop).

[Exercise: Show that, considering only the three doubles rule, the expected number of rolls per turn is 43/36.]

• After Jail, Illinois Avenue is the most landed on property. Without the Chance and Community Chest cards, Tennessee Avenue would hold this distinction. The first Chance space is the least occupied.

• The orange monopoly is the "best" investment (in the sense of shortest break-even time) among all fully improved monopolies, followed by light blue, red, maroon, dark blue, yellow, railroads, green, purple, and utilities.

• Before the appearance of any houses or hotels, a player is expected to make about \$30 per roll. Hence, a game in which no houses or hotels are purchased is likely to last a long time.

• A player on Indiana Avenue who has just rolled two doubles has the shortest expected time until landing in jail.

• After about 50 rolls (and no more than about 1200 rolls), all rental properties are expected to have been landed on at least once. The orange property group is expected to be purchased first, the purple group last.

• Analysis of a board game can demonstrate the power of Markov chain models.