A certain historical document is known to have been written by either author A or author B. Until now, it was thought equally likely to have been written by either author. Further evidence of authorship is now being sought in the way that the rare word 'upon' is used in the disputed document. The number of times a rare word occurs in a fixed length of text can be assumed to be a Poisson random variable (the parameter is the average rate of the word usage). In undisputed works of a similar length to the disputed text, author A uses 'upon' five times on average, while author B uses it once on average. The word occurs twice in the disputed text. Find the (posterior) probability that it was written by author A.