Author summary How long does it take for a treatment naive, growing bacterial colony to be able to survive exposure to a cocktail of antibiotics? En route to multidrug resistance, what order did the drugs become impotent in? Questions such as these that pertain to the emergence of a significant cell type in a growing population arise frequently. They are often investigated via mathematical modelling but biologically insightful results are challenging to obtain. Here we outline a general framework of a stochastically growing population spreading through a graph to study such questions and provide simple formulas as answers. The significant cell type appears upon the population reaching a target vertex. Due to their simplicity, the derived formulas are widely accessible and can be used to guide and develop intuition on a range of biological scenarios. We demonstrate this on several settings including: how a region where drugs cannot penetrate affects the emergence of resistance, and, the ordering of mutations that leads to drugs being ineffective.