$\mathcal{P}(i)$, then the probability of detecting $k$ bursts out of $n$ in $i$ beams is then given by the binomial distribution $$\mathcal{P}[k\textrm{ in }n;\, i\textrm{ beams}] = {n \choose k} \left[\mathcal{P}^k(i)\left(1-\mathcal{P}^{n-k}(i)\right)\right],$$ where $${n\choose k} = \frac{n!}{k! (n-k)!}$$ is the number of ways of picking $k$ unordered items from $n$ possibilities. In reality, the probability $\mathcal{P}(i)$ depends on the principal beam, since different feeds have different numbers and orientations of neighboring feeds. Accounting for this dependence is telescope specific. The computation for the FRBs detected at Parkes is given below.

All 15 published FRBs observed at Parkes are cataloged by @frbcat. Of these events, 14 were reported as single-beam detections[^3]; 2, 8, and 4 detections had their principal beam corresponding to the central, inner, and outer ring feeds respectively. One [@lorimer2007] was detected in 4 beams, with an inner-ring providing the principal detection. An additional burst (Ravi et al., in prep) was detected in two beams, again with an inner-ring principal beam. We defer an analysis of the probability of a given feed to be the principal beam to future work. In this paper, we take the principal beams for each event as given.

The aggregate probability of achieving these 16 