Predicting object by features probabilities I have the definition of an object provided as features probability.
Each object has it's own feature importance and probabilities.
For example for object "X", I have "color" feature (with the weight of 0.8) - the object can be blue in 80% of cases and black in 20% of cases. And "shape" feature (with the weight of 20%) - square in 30% and round in 70%.

I'm trying to create a "predictor", so if I'm observing something blue and round - (0.8 x 0.8) x (0.2 x 0.7) - probability for object X.


Does it make any sense mathematically?
If this method sounds reasonable enough, how should I handle really small numbers (I can have a really long vector of features, the final number will be really small)?
What you are attempting to do is so much like the Bayes decision theory; you can find the math behind it here. In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes' rule, also written as Bayes’s theorem) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person’s age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age. One of the many applications of Bayes' theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes' theorem may have different probability interpretations. With the Bayesian probability interpretation the theorem expresses how a subjective degree of belief should rationally change to account for availability of related evidence. Bayesian inference is fundamental to Bayesian statistics. 

Take a look at here and here too.