Let A and B be two random variables, let indices i and j take finite numbers of values, and let Ai and Bj be mutually independent copies of A and B. That is, all Ai and Bj together are mutually independent.
Let f(a,b) be a given function of pairs of values of A and B, and taking values 0 and 1. Consider the following event E:
f(A1, B1) = one specified value (0 or 1)
&
f(A1, B2) = one specified value (0 or 1)
&
.............................
&
f(Alast, Blast) = one specified value (0 or 1).
Event E can be described in the form of a table whose rows and columns are numbered by indices i and j, respectively, and whose entries are either 0 or 1. And E has some probability P(E).
The question is: how are probabilities P(E) related to each other for such tables of different sizes? Or, more generally: what is known about any mathematical structures resembling such things, in any artistic sense?
If this helps, let both A and B take exactly 8 values, each with the probability ⅛.
Thanks Peter for your comment, but looks like you understood my question pretty well. Imagine that you throw independently a few copies of a dice :D
Let me have two octahedral dice (they do exist): one red and one green. Let there be numbers from 1 thru 8 written on their faces. I throw the red one several times; let it be 11 times, for instance. That is, my letter i takes values from 1 thru 11. Then I throw the red dice 13 times (for instance). Then I make a table of 11*13 cells.
Now I explain what I put in the cell lying in the i-th row and j-th column: that is the number f(ai, bj), where ai means the outcome of the i-th throwing of the red dice, while bj means the outcome of the j-th throwing of the green dice.
All the information in the table taken together makes my event E.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability