in the sky, sort the gains of the beams towards that pixel in decreasing order: $[g_1({{\bm l}}),\, g_2({{\bm l}}),\, ...]$ etc. Here, $i=1$ is the principal beam by construction.

2.  The threshold fluence for detection in the $i^{\rm th}$ beam is proportional to $g^{-1}_i({{\bm l}}) $. The probability of a $n$-beam detection at pixel ${{\bm l}}$ is thus $$\begin{aligned}
    \mathcal{P}(n,{{\bm l}}) \propto & g^{\alpha}_n({{\bm l}})-g^{\alpha}_{n+1}({{\bm l}});& n<n_{\rm beam}\nonumber \\
                                      & g^{\alpha}_{n};& n=n_{\rm beam}\end{aligned}$$ where $n_{\rm beam}$ is the number of feed-horns.

3.  Marginalize over ${{\bm l}}$ to get $\mathcal{P}(n) = \int {\rm d}^2{{\bm l}}\mathcal{P}(n,{{\bm l}})$. We set the normalization to get $\sum_{i=1}^{i=n_{\rm beam}}\int {\rm d}^2{{\bm l}}\mathcal{P}(i,{{\bm l}}) = 1$, such that all probabilities computed are conditional upon a burst being detected.

4.  The probabilities for cases where a particular beam is chosen [*a priori*]{} as the principal beam can be computed by only integrating over the sky pixels that belong to the beam’s partition.

In Table 1, we present the probabilities for Parkes multiple-beam detections for far-field events. These probabilities were computed using frequency-averaged simulated beam patterns over the range 1182–1525 MHz. Corresponding probabilities for near-field events are given in the Appendix (Fig. \[fig:prob13\]).

\[tab:mbp\]

[llll]{} Feed,$\alpha$ & 1-beam & 