The probabilities in Table $1$ of that paper seem to be flawed. The paper describes how to calculate them, but they don't actually seem to be calculated that way. Angela's answer is a nice attempt at explaining the discrepancy, but a) that value should have been rounded to $0.190$, not $0.189$, and b) the other values in the table don't match that approach. A particularly clear example is the column titled "$\gt50^\circ$", in which $3$ and $1$ flares should result in a probability of
$ \frac{\binom 41+\binom 40}{2^4}=\frac5{16}\;, $
whereas the table gives $0.125=\frac2{16}$. I don't see a reasonable explanation for that value. It could be explained as the sum of the probabilities of events on both sides that are more extreme than the one observed (as opposed to events on one side that are more or equally extreme, which is what the text says), but that explanation doesn't work for the other columns. Angela's approximation gives $0.191$ in this case.
You might also want to check out this paper, which has a slightly more elaborate, though not much clearer explanation of the probabilities.