For a long time you’ve heard that there is no such thing as coincidence and that everything is Hashgachah Pratis. In fact you’ve been told that this is a basic Jewish belief. In fact there is an opposing view which saysthat how much Hashgachah Pratis you get is dependent on your righteousness. This can be taken two ways, at least for the purposes of this.
Hashgachah Pratis is directly related to your merit or h(m)=km where m and h are less than or equal to 1, or your hasgachah pratis equals k which equals 1 times your righteousness.
Hasgachah pratis increases at a rate of km or dm/dh=km or h(m) equals the integral of km or h(m)=.5km^2 but since h(0)=0 and h(1)=1 C=0 and k=2 so h(m)=k^2 where m and h are less than or equal to 1, or your hasgachah pratis is equal to the square of your relative merit.
After we have these formulas we can assess what the probability of an event being hasgachah pratis is if we know how righteous somebody is. We can do this with Bayesian reasoning. This formula can be used for many other things. For example, if somebody says he won the lottery and you know he tells the truth 999 times out of a thousand and that his chance of winning the lottery is one in a hundred thousand the odds that he is telling the truth are .9%. So our formula for the odds of Hasgacha Pratis for an event with Probability A are for
(1-A)m/((1-A)*m+A*(1-m)) or the probability of it happening with hasgachah (its nonrandom probability times the probability of hasgachah pratis) divided by that number plus the probability of it happening randomly. For the second case the probability is the same but with m^2 replacing m.