There is a debate underway about how effective Israel’s iron dome system is in protecting populated areas from missile attacks. On the pro side it is argued that somewhere between 85% to 90% of incoming missiles are destroyed. The con side argues that the proportion is much smaller, 40% or less. A large part of the difference comes from how one defines `destroy’. Perhaps a better term would be intercept. It is possible that about 90% of incoming missiles are intercepted. However, a missile once intercepted may not have its warhead disabled, making at least one of the fragments that falls to ground (in a populated area) dangerous.
While nailing down the actual numbers may be interesting, it strikes me as irrelevant. Suppose that any incoming missile has a 90% chance of being intercepted and destroyed (which is the claim of the builder of the iron dome technology). If the attacker launches N missiles and iron dome is deployed, the probability (assuming independence) not a single one making it through is (0.9)^N. Thus, the probability of at least one missile making it through the `dome’ is 1 – (0.9)^N. If N is large, this is large. For example, for N = 10, the probability that at least one missile makes its way through is at least 60% (thanks to anonymous below for correction). Thus, as long as the attacker has access to large quantities of missiles, it can be sure to get missiles through the dome.