This is a problem I’ve asked a few times in exams:

Star Wars Missile Defense System

Suppose that an anti-missile defense system has been implemented around New York to protect its inhabitants. It is estimated that if a single missile is fired, the probability of it getting through the defense system is .05. If 300 nuclear missiles are launched at New York, what is the probability that one or more missiles get through? Assume that the probability of the interception of an individual missile is independent of the fate of the other missiles.

Well some fraction of people taking this exam reason as follows: The probability of one getting through is .05. So the probability of at least one getting through should be $\colorbox[rgb]{1,1,1}{$300$}\colorbox[rgb]{1,1,1}{$\times $}\colorbox[rgb]{1,1,1}{$.05$}\colorbox[rgb]{1,1,1}{$=$}\colorbox[rgb]{1,1,1}{$15$}$. So New York has a 1500 % chance of being destroyed. Wow!

On more careful consideration it doesn’t make sense to say that 150 times out of 10, New York is destroyed. Does this mean that somehow, it’s more likely than 10 out of 10? There’s something wrong with this answer. But how do you do a calculation like this correctly? Probability can be pretty tricky to calculate even in seemingly simple cases like this.

Calculation of probability takes some getting use to. The problems that normally are tractable are ones that utilize some important concepts. The first one is independence.