[FabHK]

Hmm, a bit more involved maybe, but not that much. But your calculation sure seems short.

With S = sleeper, and L = letter L, and remembering "total probability":

   P(L) = P(L|S)P(S) + P(L|-S)P(-S), 

(where -S is not S), we have by Bayes

   P(S|L)
 = P(L|S) P(S) / P(L)
 = P(L|S) P(S) / (P(L|S)P(S) + P(L|-S)P(-S))
 = 1/6 * 1/5 / (1/6*1/5 + 1/4*4/5) 
 = 1/30 / (1/30 + 6/30) 
 = 1/7