Adjusting Probabilities When Using Alternative Cutoffs For Classification
First of all, you cannot always consider what a machine learning algorithm outputs as a "probability". Logistic regression outputs a sigmoid activation on a (0, 1) scale, but that doesn't magically make it so! 

We simply often scale things to a (0, 1) scale in ML as a measure of confidence. 

Also in your example, if the events are mutually exclusive (like classification), just think of them as "event 1" and "NOT event 1". Something like p(e1) + p(~e1) = 1. 

So when your book tells you to lower the threshold, it is simply saying that you require a smaller level of confidence to choose e1 over e2. This doesn't mean you are choosing one with smaller likelihood, you are simply making a conscious choice to adjust your precision-recall curve. 

There are other ways to combat class imbalance, but changing the threshold to be more sensitive to any indication of confidence of one class over another is certainly a way to do that.