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If the objects are circular, then the probability of them colliding is the probability that their distance is below a threshold. The difference between two independent normally distributed vectors is itself a normally distributed vector. The length of such a vector has a non-central chi-square distribution. So the problem boils down to computing the cumulative distribution of a non-central chi-squared variate. See the page on Probability of collision (two bivariate normal distributions) for more details.