How can a contingency table show that two events are independent?

"Suppose we have the following contingency table that shows the results of a survey we conducted at UMW on dominant handedness:\n\nMale Female\n\nLeft-handed 20 26\n\nRight-handed 160 208\n\nThe 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 ) = 46 / (368+46) ≈ .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 ) = 26 / 234 ≈ .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."