Some days ago I played Monopoly, a board game that is played using two dice. Afterwards I was eager to calculate the probability that a player arrives at one of the streets. I started by computing the probabilities to roll numbers. This is not as trivial as you might guess, because there are rules for doubles, e.g. $(1,1), (2,2)...(6,6).$  Later I realized that there are very good internet resources on Monopoly by J. Bewersdorff including a graphical simulation. It shows that the probability of arriving a particular street is close to uniform after many rounds. Except fields after the jail and special non-street fields. If you want to know more about Monopoly look at [1,German], Bewersdorff's book "Glück, Logik und Bluff: Mathematik im Spiel - Methoden, Ergebnisse und Grenzen" [2, German], or the bachelor thesis "Monopoly und Markow-Ketten" by Julia Tenie [3, German]. There is also a "FAQ" about probabilities of throwing dice [4, English] and an article at Wolfram's Mathworld on computing the probability of the points using several dice [5, English].

In what follows, I will address dice games using two dice with a double rule. Particularly, I will answer following questions:

• What is the probability of particular points (sum) of two dice using a double rule?
• What is expected number of points (sum) of two dice using a double rule? What is different compared to without double rule?

I use Monopoly's double rule:

1. If a player rolls a double, roll the dice again and add the new result to the previous result.
2. If a player rolls a double three times in a row, your turn is over. 

If you are only interested in the probabilities, then jump to Tables I and II. If you are interested in the theory in addition to the probabilities, then you will also find the answer of following question:

• How are the probability masses (probability mass function) defined? 

Lebesgue's decomposition shows that a probability distribution can be decomposed into absolute continuous, discrete, and singular-continuous parts. But in literature, different terminologies are used, which I want to unreval in what follows:

Two traditional classes of inverse problems are the estimation of absolute-continuous random parameters and the detection/classification of discrete random parameters by continuous random measurements. But what about the inference of mixed discrete-continuous problems? In the following, I will summarize my proposal of six classes of inverse problems and, hence, six classes of inferrers. See [1] for a table and formulars.