"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."