Smoothing the Counts Before You Classify

A Naive Bayes spam filter is trained on a labelled email corpus built from a vocabulary of exactly $V = 20$ distinct words.

Across the whole training set:

• The Spam class contains $N_{\text{Spam}} = 180$ word tokens in total, and $P(\text{Spam}) = 0.30$.
• The Ham class contains $N_{\text{Ham}} = 230$ word tokens in total, and $P(\text{Ham}) = 0.70$.

The raw training counts of three particular words are:

The filter uses add-one (Laplace) smoothing, so every word probability is estimated as

$P(w \mid c) = \frac{\text{count}(w, c) + 1}{N_c + V}$

A new message arrives containing exactly the three words offer, meeting and click, each once. The classifier assumes the words are conditionally independent given the class.

What is $P(\text{Spam} \mid \textit{offer}, \textit{meeting}, \textit{click})$?

Give your answer as a percentage, rounded to 2 decimal places.