More generally, that the chance of each dog X failing to detect each mine Y is independent. If it is independent, then you get the (1-P)^3 chance of failure, m

More generally, that the chance of each dog X failing to detect each mine Y is independent. If it is independent, then you get the (1-P)^3 chance of failure, meaning 0.1% for all three failures if each dog has a 90% chance of succeeding. But of course that isn't true in reality. There will be properties of the mines (burial depth, ground cover, degree of leakage of sniffable explosive material) that affect its chance of being found by all dogs searching for a mine. If you wanted the parent poster's statistical chance, you'd want three 90% ways of detecting mines that are truly independent. Perhaps a dog, and infrared imaging sensitive enough to detect the explosive material (I have no idea if that's possible), and this article's mechanism.