[L]et's say you almost get run over by a car[; t]here was a very real probability that you may have been runover […], but you didn't. Why isn't there some civil repercussions for that?
Construing the first question broadly, and in line with the one reading "[w]hy isn't there some civil repercussions for [a missed although very real probability of a car run-over]?", the question is, admittedly, based on a few incorrect premises. There are "repercussion" some of them being of criminal nature (reckless driving, vehicular assault etc.), and others may become of civil nature as follows:
Common law tort of negligence — Mere threat of harm can be harm for purposes of damages
"“When there is a breach of duty, "a person who is in the path of negligent conduct and reasonably fears for his or her own safety may recover for resulting emotional distress." In re Air Crash Disaster Near Cerritos, Cal., 973 F.2d 1490, 1493 (9th Cir. 1992). ” […] See Potter, 863 P.2d at 833 (George, J., concurring and dissenting) (discussing hypothetical pedestrian narrowly avoiding speeding car and indicating that threat of injury is the relevant issue); Wooden v. Raveling, 71 Cal. Rptr. 2d 891, 897-98 (Cal. Ct. App. 1998) (quoting Potter hypothetical and holding plaintiff was not precluded from relief simply because car did not actually hit her) (Taylor v. Honeywell Int'l, Inc. (9th Cir. 2015) 599 F. App'x 664, 2) (bold type added)
Accordingly, broadly construing the question so as to effect the greatest scope of the spotting of damages, one may sue, although not for injuries per se, but instead damages under an emotional distress legal theory and its particular categories like anxiety, depression, recurring nightmares, sleeplessness, anger, angst etc. depending on the actual facts of the case, and typically supported by expert testimony when possible.
Strictly construing the question, the mere possibility or even probability of injuries do not merit a cause of action for damages in and of themselves, as such damages never occurred.
It may be possible that in certain scenarios the mere possibility is so outrageous that it exceeds all bounds of a civilized society, that nominal damages, say, of $1 are awarded and punitive damages are awarded so as to deter such conduct, but even in that case such a hypothetical case the cause of action will not be the damages under a personal injury tort, but one for nominal damages and for punitive damages even if the factual underpinnings are the same.
Standing is one's positive relation to at least one event that is the basis of a legal proceeding in a court, that is, one being the presumptive subject of some sort of wrongdoing that merits their recognition as a party to a legal proceeding. The term there would have been "damages" instead of standing.
Why doesn't law take into account probability?
It does. Rarely enumerated probabilities other than in case of the preponderance of the evidence standard of proof where the quality of the evidence must weigh in favor of the one who has the burden of proof, in other words the fact finders job is to decide whether something is greater or smaller than 50 percent probability.
Although it is rather the exception than the rule, probabilities other than the greater-than-50-percent standard also appear here an there in the judicial process, for example, in asylum cases “"[t]o effect a well-founded fear [from persecution so as will constitute protected for purposes of U.S. asylee or refugee status], a threat need not be statistically more than fifty-percent likely [to have been made; however,] the [U.S.] Supreme Court has suggested that even a one-tenth possibility of persecution might effect a well-founded fear." Lim,224 F.3d at 934-935 (citing INS v. Cardoza-Fonseca,480 U.S. 421, 430, 107 S.Ct. 1207, 94 L.Ed.2d 434 (1987)). Kaiser v. Ashcroft, 390 F.3d 653, 658 (9th Cir. 2004) (see also INS v. Cardoza-Fonseca, [“ "Let us ... presume that it is known that in the applicant's country of origin every tenth adult male person is either put to death or sent to some remote labor camp.... In such a case it would be only too apparent that anyone who has managed to escape from the country in question will have 'well-founded fear of being persecuted' upon his eventual return”])
In less enumerated forms many other aspects of the judicial process rely on probabilities.