The authors, an interdisciplinary team of three mathematicians and two historians, examine four episodes in military history: the battles of Jutland 1916 and of Britain 1940, Vietnam 1965–73 and nuclear deterrence 1945–83. In each case they provide historical context and ask a counterfactual question: Could the Germans have won Jutland? Or won the Battle of Britain? Could the US have won in Vietnam? Could the 1983 Able Archer exercise have prompted nuclear war? In each case they adopt mathematical procedures, Lanchester Laws, Approximate Bayesian Computation, bootstrapping, data analysis and game theory to consider ‘restrained’ rather than ‘exuberant’ counterfactuals. Restrained counterfactuals make small changes, where the consequences, particularly the role of luck, can be quantified. Some of the questions are quite precise. For instance, they ask what is the probability that specified changes in German strategy could have reduced British pilot strength below 1,000 in 1940? Since they have data on losses when those strategies were being followed, the estimates are plausible. Other questions, like the probability of nuclear war, are more speculative and they compare the movement of their estimated probabilities with the time on the Bulletin of the Atomic Scientists Doomsday Clock. The procedures are explained in a non-technical way, without equations. They have an appendix titled ‘Some mathematical background (with no equations)’. Their calculations rest on more technical work and they provide references to these sources, which do have the equations. They note that they are struck, despite their intentions, by the extent to which personalities matter. Their calculations illuminate the individuals who made the crucial decisions, which could have been made differently.