Can we use mathematics and logic to estimate probability of extremely absurd events?

Question

I'd like to detail my question over the example below.

Let's say I have a random pixel generator which has $1024 \times 768$ screen resolution. It also has $24$ bit color which means $2^{24}= 16,777,216$ possible colors per pixel. Therefore, for each pixel on the screen we have a $1$ in $1.319414e+13$ using $(1024 \times 768\times 16,777,21)$ chance of being set at the correct position to generate any image we might think of.

Let's say I want to calculate probability of generating the image of Mona Lisa painting which is of course extremely unlikely to happen yet mathematically possible. This is indeed another version of famous Infinite Monkey Theorem where we deal with unfathomably large numbers that are hard to comprehend in real life.

The chance of seeing Mona Lisa painting by randomly shuffling the pixels is therefore $1$ in $1.319414e+13$ which is pretty straightforward calculation that can be simply deduced by using our mathematical knowledge.

So far so good but let's make things more absurd. In another scenario Mona Lisa painting is coming to my mind hence I am imagining the portrait and pressing the shuffle button. What are the chances of having exact same image that I imagined on the random pixel generator?

Should I treat my imagination of Mona Lisa and having Mona Lisa on the screen as two different independent events i.e $P(A)$ & $P(B)$? As such can I make similar pixel calculation for the image of Mona Lisa in my mind treating it as another random pixel generator and calculate likelihood of having Mona Lisa on the screen just after imagining it as $P(A)\cdot P(B)$?

Are rules of probability theory sufficient to calculate likelihood of such bizarre event (no matter how rough estimations we can do)? If my deduction is inaccurate what kind of methodology should one follow to come up with an estimation?

Answer 1

Can we use mathematics and logic to estimate probability of extremely absurd events?

We can estimate the probability of anything that has certain properties, regardless of whether it is absurd or not. One of these properties is a defined sample space. That something is absurd has no relation to the properties required to estimate its probability.

Should I treat my imagination of Mona Lisa and having Mona Lisa on the screen as two different independent events i.e P(A) & P(B)?

Yes. The fact that you imagine Mona Lisa has no influence on whether it appears on the screen.

can I make similar pixel calculation for the image of Mona Lisa in my mind treating it as another random pixel generator and calculate likelihood of having Mona Lisa on the screen just after imagining it as P(A) * P(B)?

Treating your imagination like a pixel generator is absurd and arbitrary. I don´t see how such decision could be justified nor, even if it was, what the criteria would be for deciding the properties of such "pixel generator".

Are rules of probability theory sufficient to calculate likelihood of such bizarre event (no matter how rough estimations we can do)?

No. It is impossible to define a sample space $\Omega$, a proper measurable function $f:\Omega \rightarrow Y$, or any other property relevant to probability theory for a concept such as 'your imagination' (whatever 'imagination' means).