A study reported by Forbes indicated that the five most common words appearing in spam e-mails are shipping!, today!, here!, available, and fingertips! Many spam filters separate spam from ham (e-mail not considered to be spam) through application of Bayes' theorem. Suppose that for one e-mail account, 1 in every 10 messages is spam and the proportions of spam messages that have the five most common words in spam e-mail are given below.

Shipping! ..................051

Today! ......................045

Here! ........................034

Available ..................014

Fingertips! ...............014

Also suppose that the proportions of ham messages that have these words are

Shipping! .............0015

Today! .................0022

Here! ...................0022

Available ............0041

Fingertips! .........0011

a. If a message includes the word shipping!, what is the probability the message is spam? If a message includes the word shipping!, what is the probability the message is ham? Should messages that include the word shipping! be flagged as spam?

b. If a message includes the word today!, what is the probability the message is spam? If a message includes the word here!, what is the probability the message is spam? Which of these two words is a stronger indicator that a message is spam? Why?

c. If a message includes the word available, what is the probability the message is spam? If a message includes the word fingertips!, what is the probability the message is spam? Which of these two words is a stronger indicator that a message is spam? Why?

d. What insights do the results of parts ( b) and (c) yield about what enables a spam filter that uses Bayes' theorem to work effectively?