  The problem addressed concerns the determination of the average number of
successive attempts of guessing a word of a certain length consisting of
letters with given probabilities of occurrence. Both first- and second-order
approximations to a natural language are considered. The guessing strategy used
is guessing words in decreasing order of probability. When word and alphabet
sizes are large, approximations are necessary in order to estimate the number
of guesses. Several kinds of approximations are discussed demonstrating
moderate requirements concerning both memory and CPU time. When considering
realistic sizes of alphabets and words (100) the number of guesses can be
estimated within minutes with reasonable accuracy (a few percent). For many
probability distributions the density of the logarithm of probability products
is close to a normal distribution. For those cases it is possible to derive an
analytical expression for the average number of guesses. The proportion of
guesses needed on average compared to the total number decreases almost
exponentially with the word length. The leading term in an asymptotic expansion
can be used to estimate the number of guesses for large word lengths.
Comparisons with analytical lower bounds and entropy expressions are also
provided.
