Second, Introverted Thinking judges the environment according to how things are rationally reasoned internally and judges the environment based on internally *logically* reasoned principles and conclusions (it does not *overwrite* the environment). If 5 + 5 = 10, and there are 5 boxes being put next to 5 other boxes in the environment, how many boxes are there now? With Ti, the mathematical model of 5 + 5 is applied to the sensory observation (5 boxes are being put next to 5 other boxes), reasoning that since 5 + 5 = 10, and 5 same entities are being put next to 5 of the same entities, then there are 10 of the same entities in the environment. Of course it doesn't work this clear cut for everything, as Ti works fast and people aren't stupid and can figure something out like that easily. A better example would be something much more complex. Let's say that you don't remember the surface area formula for a cube, and you need to figure it out with the given Edge = 5 cm. First, you try to figure out what the concept of Surface Area is (logical analysis), the total area on the surface of a 3-dimensional figure (in this case). Reasoning that a Cube has 6 faces that are each 2-dimensional squares, you logically conclude that if you add the areas of the 6 squares, then you will also have the entire surface area of the Cube, 150 sq cm. You have observed the problem, analyzed it, and judged (because projected apparently doesn't compute) internally *logically* reasoned knowledge onto the environment (the algorithm, that the 6 squares' area will equal the cube's surface area) to solve the problem that has been observed. Congratulations, you have just applied/judged/projected a principle/concept onto the environment to solve a problem/mystery.
Because I'm psychic and somehow know that you have read the above and thought "but what does Te do then" (I at least hope you thought of that, because if you are still stuck on the whole projection = judging I'm somewhat disappointed). Thinking as a function is responsible for logical analysis and judging. The Te approach to the problem would either be based off the Si recollection of previous methods or the Ni realizations of the circumstances producing a general idea to solve surface area that have been gathered from an external source (Te, adopting external logical frameworks/knowledge). Another approach would be identical to that of the Ti function's reasoning in the sense that the Te-user will organize the environment and use examples to prove points that will lead to the answer, as well as logically laying out the steps as to how to reach the solution in a clear-cut fashion, usually documenting details. The Te user would have an easier time visualizing or actually having a hollow cube at their disposal to break apart into 6 separate squares (so would Se types, which only adds to the confusion), proving to the Te user that the surface area of the cube (surface area's analytic meaning being reasoned out from prior knowledge or through unconscious Ti use, everyone uses the function to some extent) is the exact same as the area of the 6 squares. Furthermore, the Te user will often work through problems such as these externally, working in a Ti-ish fashion except entirely external, usually with either writing down and organizing information or logically engaging in an explanation to another party that can offer more information and allow the Te user to logically work through the problem via explanation.
Also, for the facial expression thing, pretty much everyone without specific deficits as seen in disorders such as ASD will do fairly well in extracting the emotional implications from others' faces. The Fe-Pi mechanism is more about extracting other's attachments to certain things and interpreting the emotional affect objects in the environment have on others (objects being both animate and inanimate), as well as adopting external ethical models from the environment to be interpreted and subsequently applied.
Use of probabilities is the domain of both Thinking functions (and I like to call Ni zeroing "probability analysis", but it is more the extraction of meaning from probabilities gathered from an external source). The difference in that argument is that the Fi type is likely to come to its own conclusions based on ethical reasoning (people with goatees are evil because all the ones I've ever seen hurt people) judging the people with goatees with the concept of evil (but, this, of course, also usually leads to the conclusion, "well, I haven't met every single person with a goatee, I've only met the ones I've met; therefore it is still possible that there are ethically good people with goatees", which is a more Ne perspective and even the dichotomial Perceiving argument to keep options and possibilities open), and that the Te type is more likely to appeal to credible empirical studies conducted by others to interpret the data and judge the environment based on the interpretation, and from these people, you can expect "studies show" kind of arguments that appeal to other frameworks that have been adopted rather than created internally as is seen in introverted judging processing."Overall, I take it you were trying to say that the Te user just attributes a probability to people with goatees hurting other people while the Fi (and Ti??) user is thinking black-and-white along the lines of "they will always hurt people"??? That makes no sense to me, why couldn't the JiPe user use probabilities? Staying with Ti now, probability theory is a pretty pure logic thing :P"