Accidental coincidence ($R_{\mathrm{acc}}\simeq r_1{r}_{2}w$)

Suppose two independent background streams follow Poisson processes with rates $r_1$ and $r_2$. Define a coincidence by $|t_2-t_1|<w/2$: $w$ is the full accepted time-difference width. Around each first-counter signal, the expected number of second-counter signals is $r_{2}w$. The expected pair rate is therefore $r_1{r}_{2}w$. A circuit producing at most one output per first signal instead givesThe distinction matters when multiple detections occur in one window. If a different convention accepts $|t_2-t_1|<\tau$, its full width is $w=2\tau$ and the dilute expression is $2r_1{r}_2\tau$.

For a chosen numerical model, take background rates $r_1=100{\mathrm{s}}^{-1}$, $r_2=80{\mathrm{s}}^{-1}$ and $w=20\mu \mathrm{s}$. The dilute pair rate is $0.16{\mathrm{s}}^{-1}$. These are stipulated teaching inputs, not measurements of the historical chamber.

Making $w$ smaller also rejects some genuine events. Suppose true pairs arrive at rate $S_0$ before the timing cut, and their time difference has a centered normal distribution with standard deviation ${\sigma }_t$. Its accepted fraction, using the error function, isFor chosen $S_0=2.0{\mathrm{s}}^{-1}$ and ${\sigma }_t=5\mu \mathrm{s}$, the $20\mu \mathrm{s}$ window retains about $95.45\%$ of true pairs. In the dilute background model the fraction of accepted events that are true is $S/(S+R_{\mathrm{acc}})\simeq 92.27\%$. A narrow window trades detection efficiency against background; it does not make either quantity equal to one. Delaying one counter's signal far beyond genuine event correlations gives a proposed background check under stationarity, without assuming that the zero-delay excess itself is accidental.