Your terminology is not standard ("scale by a known weight") so it's not clear how an outcome might be modified in a probabilistic manner. One specific that needs clarification is "Do the modified outcomes still only consist of the integer values from 0 to 8?"
If so, then one possibility is that one might consider
$p$ to be a random variable rather than the constant 0.22. One way to do this is the following:
(* Modify the logit of the probability by adding some noise *)
s = 0.05;
SeedRandom[12345];
logitp = Log[0.22/(1 - 0.22)] + RandomVariate[NormalDistribution[0, s], 1][[1]]
(* -1.3151 *)
(* Transform back to a probability *)
p = 1 - 1/(1 + Exp[logitp])
(* 0.211634 *)
(* Generate a random observation *)
x = RandomVariate[BinomialDistribution[8, p], 1][[1]]
(* 1 *)
There are lots of other ways, too. Hence, the need for more specifics.