A study by Forbes indicated that the five most common words appearing in spam emails are shipping!, today!, here!, available, and fingertips!. Many spam filters separate spam from ham (email not considered to be spam) through application of Bayes' theorem. Suppose that for one email account, 1 in every 10 messages is spam and the proportions of spam messages that have the five most common words in spam email are given below. shipping! 0.051 today! 0.045 here! 0.034 available 0.014 fingertips! 0.014 Also suppose that the proportions of ham messages that have these words are: shipping! 0.0015 today! 0.0022 here! 0.0022 available 0.0041 fingertips! 0.0011 a. If a message includes the word shipping!, what is the percent probability the message is spam? If a message includes the word shipping!, what is the percent 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 percent probability the message is spam? If a message includes the word here!, what is the percent probability the message is spam? Which of these two words is a stronger indicator that a message is spam? Why? c. If a messages includes the word available, what is the percent probability the message is spam? If a message includes the word fingertips!, what is the percent probability the message is spam? Which of these two words is a stronger indicator that a message is spam? Why?

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