Infinity is not a number.

I know that there are some algebra that included infinity as possible values. For example, in homogeneous coordinates, or when computing cross-ratio.

But even there, infinity is not a number. If you do computation in a set than contains infinity as if it were a normal value, even then, infinity is not a number; the set of possibles values contains numbers and infinity that's all.

This is maths. So everything is possible if you want it to be. If you want to say "I decide to consider infinity as a number", you can. But if you do, you need to compute all the rules of your new aximoatic systems. And then you can't in that axiomatic system of yours use notion such as "prime number" (or you also need to define them; and then, it is up to you to decide if infinity is prime)

For example, if infiity is a number, then what is infinity + 1?

You have to decide that (it is a convention nothing else).

If you say "you can't add a number to infinity", then you can't neither say it is a prime.

If you say "infinity+1=infinity", then infinity is both odd and even. Which, I guess, can't make it a prime. At least not with any definition of "prime" that ressembles the one we know in the "normal" aximatic system.

So, in short, unless you want to redefine a whole axiomatic system (in which case, it is us who should ask you whether infinity is prime or not), infinity is not a number. And therefore not a prime number.

All that being said, if you really define an aximatic system where infinity is a number, and where infinity has a primality, then I would bet that infinity is not a prime.

Because probability of being prime decrease when the number increases.

The bigger x is, the less likely x is to be prime.

Exactly like, when you use extensions of set of numbers that includes infinity, you usually defines 1/infinity as 0 (because the bigger x is, the smaller, in absolute value 1/x is; so when x is bigger than any number, that is when x is infinity, 1/x has to be smaller than any number in absolute value, that is 0)

Likewise here, I would therefore say that the probability that infinity is prime is equal to 0.

But

1) Good luck to define a coherent axiomatic system where this makes sense.

2) Even then, "probability of something to be prime equals 0" is not the same as "that something is not prime". I know that common sense dictate that if probability is 0, then it is impossible. But it is not true among infinite outcomes. Probability of a number to be integer is also 0. Yet 1, 2, or 3 are integers. So infinity could very well be prime without contradicting with the probabilistic consideration.