The occurrence frequency of cirrus clouds as function of ambient relative humidity over ice, based on in-situ observations performed during the INCA experiment, show a clear difference between the campaign carried out at Southern Hemisphere (SH) midlatitudes and the campaign carried out at Northern Hemisphere (NH) midlatitudes. At a given relative humidity above ice saturation, clouds are more frequent in the NH. At relative humidities near ice saturation, clouds defined as containing particles with sizes larger than 0.55 µm diameter and an integral number density above 0.2 cm^-3 were present 70% of the time during the SH campaign, whereas clouds where present 95% of the time during the NH campaign. Using a size threshold of 1 µm diameter to define the presence of clouds result in a less frequent occurrence of 60% of the time in the SH campaign and 75% of the time in the NH campaign. The data show that the presence of particles is a common characteristic of cirrus clouds. Clouds at ice saturation defined as having crystal sizes of at least 5 µm diameter and a number density exceeding 0.001 cm^-3 were present in about 80% of the time during the SH campaign, and almost 90% of the time during the NH campaign. The observations reveal a significant cloud presence fraction at humidities well below ice saturation. Local minima in the cloud presence fraction as a function of relative humidity are interpreted as systematic underestimation of cloud presence because cloud particles may become invisible to cloud probes. Based on this interpretation the data suggests that clouds in the SH form preferentially at relative humidities between 140 and 155%, whereas clouds in the NH formed at relative humidities less than 130%. A simple assumption about the probability to reach successively higher humidities in an ice supersaturated air parcel provides a model that explains the main trend of the cloud presence fraction as function of relative humidity. If adiabatic processes are assumed a cloud water content distribution can be derived from this probability model. The resulting distribution agrees well in shape compared to observations, but the observed mean cloud water content is less than expected from simply adiabatic processes.