$r=M_B/5.2895$, and the $p_i$ are parameters given by the fit.

We next view the $M_B$ distribution for events having $\Delta E$ within 2$\sigma$ around zero in Fig. \[bm\_4pi\_kpi\](b). This distribution is fit with a Gaussian signal function of width 2.7 MeV and the background function found above whose normalization is allowed to vary. We find 358$\pm$29 events in the signal peak.

We repeat this procedure for the other two $D^o$ decay modes. The $M_B$ spectrum for $\Delta E$ sidebands and signal region is shown in Fig. \[bm\_4pi\_k2pi\]. The $\Delta E$ resolution is 22 MeV in the $K^-\pi^+\pi^o$ mode and 18 MeV in the $K^-\pi^+\pi^+\pi^-$ mode. Signal to background ratios are worse in these two modes, but the significance is still large. The number of signal events in each mode are shown in Table \[table:4pievents\].

  $D^o$ Decay Mode        Fitted \# of events (%)
  ---------------------- -------------------------
  $K^-\pi^+$                    358$\pm$29
  $K^-\pi^+\pi^o$               543$\pm$49
  $K^-\pi^+\pi^+\pi^-$          329$\pm$41

  : Event numbers for the $D^{*+}\pi^+\pi^-\pi^-\pi^o$ final state[]{data-label="table:4pievents"}

We choose to determine the branching fraction using only the $D^o\to K^-\pi^+$ decay mode because of the relatively large backgrounds in the other modes and the decreased systematic error due to having fewer particles in the final state. In order to find 