Atmospheric cloud condensation nuclei (CCN) concentrations are a key uncertainty in the assessment of the effect of anthropogenic aerosols on clouds and climate. The ability of new ultrafine particles to grow to become CCN varies throughout the atmosphere and must be understood in order to understand CCN formation. We have developed the Probability of Ultrafine particle Growth (PUG) model to answer questions regarding which growth and sink mechanisms control this growth, how the growth varies between different parts of the atmosphere and how uncertainties with respect to the magnitude and size distribution of ultrafine emissions translates into uncertainty in CCN generation. It was found in most cases that condensation is the dominant growth mechanism and coagulation with larger particles is the dominant sink mechanism for ultrafine particles. In this work we found that the probability of a new ultrafine particle generating a CCN varies from <0.1% to >90% in different parts of the atmosphere, though in the boundary layer a large fraction of ultrafine particles have a probability between 5% and 40%. Some regions, such as the tropical free troposphere, are areas with high probabilities; however, variability within regions makes it difficult to predict which regions of the atmosphere are most efficient for generating CCN from ultrafine particles. For a given mass of primary ultrafine aerosol, an uncertainty of a factor of two in the modal diameter can lead to an uncertainty in the number of CCN generated as high as a factor for eight. It was found that no single moment of the primary aerosol size distribution, such as total mass or number, is a robust predictor of the number of CCN ultimately generated. Therefore, a complete description of the size distribution is generally required for global aerosol microphysics models.