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Everyone loves the union bound. It is the most dependable of all probability bounds. No matter what mess of dependencies threaten your analysis, the union bound will ride in and save the day. Except when it doesn’t.
I’ve been giving a talk about some recent results (here are some slides) and have received a particular question from a couple of different audiences. The exact details aren’t important but the basic gist was why I didn’t simplify some of the analysis by just applying the union bound. The short answer was it didn’t apply. But how could that be? My longer answer was longer and probably less helpful. So let me distill things down to the following paradox (which isn’t really a paradox of course).
The Geography Test. Suppose you have a geography test tomorrow. In the test you’ll be asked two questions out of five possible questions (and you already know the list of possible questions). If you get both right, you’ll pass the course. However, rather than revise everything you adopt a random strategy that ensures that for any question, you’ll have a correct answer with probability at least 4/5. Hence, by the union bound, you’ll get both questions right with probability at least 3/5.
Right? Sure, it’s a straight-forward application of the union bound: the probability of being wrong on one or both questions is at most the probability of being wrong on the first (at most 1/5) plus the probability of being wrong on the second (at most 1/5). And the union bound never let’s you down.
The Bad News. You’re going to fail both your geography and probability classes. Consider the following scenario. The five possible exam questions are: 1) name a city in the US, 2) name a city in France, 3) name a city in Germany, 4) name a city in the UK, and 5) name a city in Italy. Your cunning strategy is to memorize one of the rows in the following table.
You choose the row to remember uniformly at random thus guaranteeing that you can answer any question correctly with probability 4/5. Of course the professor doesn’t know which row you’ll pick but he does know your strategy. (This is equivalent to the usual assumption we make about adversaries knowing your algorithm but not your random bits.)
Unfortunately, you stay up so late studying your row that you sleep in and miss the exam. You go to the professor’s office to plead for a make-up exam. He is initially reluctant but then a devious smile crosses his face. Sure, he’ll still give you the exam. You’re relieved that you’ll still pass the course with probability at least 3/5. The professor doesn’t look so sure.
The first question he asks is to name a city in the US. “Philadelphia” you answer with confidence. As the professor nods you see that devious smile again. The second question… name a city in the UK.
And then you realize that the probability of success was never as large as 3/5 since the professor could always guarantee that you fail by first asking for a US city and then choosing his second question accordingly.
Morals of the story: Don’t miss exams, study hard, and don’t blindly trust the union bound.