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This is a typeset version of Alan Turing's Second World War research paper \textit{The Applications of Probability to Cryptography}. A companion paper \textit{Paper on Statistics of Repetitions} is also available in typeset form from arXiv at arXiv:1505.04715. The original papers give a text along with figures and tables. They provide a fascinating insight into the preparation of the manuscripts, as well as the style of writing at a time when typographical errors were corrected by hand, and mathematical expression handwritten into spaces left in the text. Working with the papers in their original format provides some challenges, so they have been typeset for easier reading and access.

The Applications of Probability to Cryptography demonstrates that Turing was determined to apply rigorous probability analysis to a wide range of cryptanalytic problems of the day. A particular highlight is where Turing uses life expectancy to examine conditional probability. The associated example, Hitler is now of age 52 , adds piquancy and suggests that the paper was written between April 1941 and April 1942. Bletchley Park's output of decrypts was almost certainly enabled by the techniques in this paper.