This was closed as off-topic on math.se, and it was suggested I post this here, so here goes.
Firstly, I am aware that this thread exists, and I'll definitely be ordering a copy of the book, "Math on Trial...", in the top answer. My question, while related, is more specific.
I imagine there are many, many examples of the normal distribution being useful in decision-making of processes, such as whether or not to modify a bolt-making factory machine depending on whether it passes quality control standards, and this is found out by taking a sample of bolts and approximating the mean and variance of the length of the bolt, and seeing if the mean is off or if the variance is too large.
Another example could be in medicine: if the gaps between heart beats follows some sort of modified normal distribution; that could tell doctors if someone's heart-rate variability is high enough so that medical intervention is required.
But my question here today is about how statistical distributions - Binomial, Normal, Poisson or otherwise was critical in the decision of the ruling of a law, be it in a court case or otherwise.
Something I have in mind is if an individual, or a gambling site/entity, was accused of cheating and a hypothesis test with the normal distribution was a key factor in a trial. If it was the individual accused of cheating, then maybe the evidence was that it was that their winnings were statistically too unlikely to be coincidental. On the other hand, if it was the gambling company accused of being rigged, then maybe statistical evidence was found that that the probabilities in whatever game was played, was found to not be equal to the probabilities if the game were truly fair.
But also, it doesn't have to be a court case: it could be how calculations from statistical distributions affected government policy (law), e.g. decision making when it came to the placement of speed cameras on a road or the height of a building or whatever decision that policy-makers have to make.