The applet illustrates the probability distribution of Gaussian noise around a signal and the relation of triggering probability to signal, rms noise, and threshold levels.
The magenta curve illustrates the basic signal + noise probability distribution. Markers show the ±1σ rms noise extent. The curve width and height are governed by the rms noise level, and the mean x-axis position is governed by the signal level; while the area below the magenta curve and to the right of the threshold indicator represents the threshold triggering probability
The blue curve is the inverse of the cumulative amplitude probability and where this crosses the threshold line is the probability of triggering and signal detection.

User Notes:
The total area under the probability distribution curve is the maximum cumulative probability of detection = 1.
The cumulative probability, or the probability that signal + noise exceeds the threshold = 0.5 occurs when the threshold and signal levels are equal.
Raising the signal level via the scrollbar, relative to the threshold setting indicates that the threshold triggering probability at any instant is still finite, even when the signal is less than the threshold value.
Also note that threshold triggering is not certain, even when the mean signal level exceeds the threshold setting by a moderate amount.
This region of uncertainty is threshold-noise-ratio dependent and can be explored using the rms Noise and Threshold scrollbars.