4.7: Independence

We’ve seen that a particular problem can involve multiple different events. In the All-time Idol example, we considered the probability of a female winner, a country singer winner, and an underage winner, among other things.

Now one question that often arises concerns the independence of events. Two events \(A\) and \(B\) are called independent if the prior probability is the same as the conditional probability; that is, if Pr(\(A|B\)) = Pr(\(A\)).

If you reflect on what this means, you’ll see that with independent events, knowing that one of them occurred tells you nothing (either for or against) about whether the other one also occurred.

For example, let \(S\) be the event that Strike For Gold wins the Kentucky Derby next May. Let \(R\) be the event that it rains that day. If I say that \(S\) and \(R\) are independent, I’m claiming that rain (or the absence thereof) would have no impact either way on the horse’s chances. If you were able to see the future, and reveal to me the weather on Derby Day, that’s fine but it wouldn’t help me in my betting. Knowing Pr(\(R\)) wouldn’t give me any helpful information, because Pr(\(S|R\)) is the same as just plain old Pr(\(S\)) anyway.

That’s a conceptual explanation. In the end, it boils down to numbers. Suppose we have the following contingency table that shows the results of a survey we conducted at UMW on dominant handedness:

	Male	Female
Left-handed	20	26
Right-handed	160	208

The data is self-explanatory. Obviously there were a lot more right-handers who took our survey than left, and slightly more women than men. Now consider: if this data is reflective of the population as a whole, what’s Pr(\(L\)), where \(L\) is the event that a randomly chosen person is left-handed? We surveyed 160+208=368 right-handers and only 20+26=46 southpaws, so we’ll estimate that Pr(\(L\)) = \(\frac{46}{368+46} \approx\) .111. If you pick a random person on campus, our best guess is that there’s a .111 probability of them being left-handed.

Suppose I told you, however, before you knew anything about the randomly chosen person’s handedness, that she was a woman. Would that influence your guess? In this case, you’d have extra information that the \(F\) event had occurred (\(F\) being the event of a female selection), and so you want to revise your estimate as Pr(\(L|F\)). Considering only the women, then, you compute Pr(\(L|F\)) = \(\frac{26}{234} \approx .111\) from the data in the table.

Wait a minute. That’s exactly what we had before. Learning that we had chosen a woman told us nothing useful about her handedness. That’s what we mean by saying that the \(L\) and \(F\) events are independent of each other.

The shrewd reader may object that this was a startling coincidence: the numbers worked out exactly perfectly to produce this result. The proportion of left-handed females was precisely the same as that of left-handed males, down to the penny. Is this really likely to occur in practice? And if not, isn’t independence so theoretical as to be irrelevant?

There are two ways of answering that question. The first is to admit that in real life, of course, we’re bound to get some noise in our data, just because the sample is finite and there are random fluctuations in who we happened to survey. For the same reason, if we flipped an ordinary coin 1,000 times, we aren’t likely to get exactly 500 heads. But that doesn’t mean we should rush to the conclusion that the coin is biased. Statisticians have sophisticated ways of answering this question by computing how much the experimental data needs to deviate from what we’d expect before we raise a red flag. Suffice to say here that even if the contingency table we collect isn’t picture perfect, we may still conclude that two events are independent if they’re “close enough" to independence.

The other response, though, is that yes, the burden of proof is indeed on independence, rather than on non-independence. In other words, we shouldn’t start by cavalierly assuming all the events we’re considering are in fact independent, and only changing our mind if we see unexpected correlations between them. Instead, we should always be suspicious that two events will affect each other in some way, and only conclude they’re independent if the data we collect works out more or less “evenly" as in the example above. To say that Pr(\(A|B\)) is the same as Pr(\(A\)) is an aggressive statement, outside the norm, and we shouldn’t assume it without strong evidence.

One more point on the topic of independence: please don’t make the mistake that mutually exclusive events are independent! This is by no means the case, and in fact, the opposite is true. If two events are mutually exclusive, they are extremely dependent on each other! Consider the most trivial case: I choose a random person on campus, and define \(M\) as the event that they’re male, and \(F\) as the event that they’re female. Clearly these events are mutually exclusive. But are they independent? Of course not! Think about it: if I told you a person was male, would that tell you anything about whether they were female? Duh. In a mutual exclusive case like this, event \(M\) completely rules out \(F\) (and vice versa), which means that although Pr(\(M\)) might be .435, Pr(\(M|F\)) is a big fat zero. Pr(\(A|B\)) is most certainly not going to be equal to Pr(\(A\)) if the two events are mutually exclusive, because learning about one event tells you everything about the other.