If the number of the bits were to be infinite, then there would be an isomorphism from the board into the Rado graph, and since it contains every finite graph, then the probability of the board containing all the possible finite computation would be 1.
The number of bits contained in our cosmological horizon is much much less then the number of bits contained in 3->3->3 cells, and since Life is TC, I suspect that similarly the probability of such a board of containing an AI is uncalculably close to 1.
I suspect also that you could make 1 - P(AI in life) as small as you want by adjusting the size of the board.
There are three very important distinction between "bits contained in our cosmological horizon" and "number of bits contained in 3->3->3 cells" of Conway's life: 2D vs 3D, the complexity of the "rules" governing interaction between the "bits" and the number of possible states of each "bit". This almost certainly (as in "I am very sure, but a formal proof eludes my math skills") means that 3->3->3 cells is far too few to simulate even a small portion of our universe.
I tried for a couple...
Conway’s Game of Life is Turing-complete. Therefore, it is possible to create an AI in it. If you created a 3^^3 by 3^^3 Life board, setting the initial state at random, presumably somewhere an AI would be created. Would this AI somehow take over the whole game board, if given enough time?
Would this be visible from the top, as it were?
EDIT: I probably meant 3^^^3, sorry. Also, by generating at random, I meant 50% chance on. But any other chance would work too, I suspect.