Refer to Example 31 concerning spam but now suppose that among the 5000 messages, the 1750 spam messages have 1570 that contain the words on a new list and that the 3250 normal messages have 300 that contain the words.

(a) Find the probability that a message is spam given that the message contains words on the new list.

(b) Would you prefer the new list here or the one in Example 31? Why?

Data From Example 31

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EXAMPLE 31 Identifying spam using Bayes' Theorem A first step towards identifying spam is to create a list of words that are more likely to appear in spam than in normal messages. For instance, words like buy or the brand name of an enhancement drug are more likely to occur in spam messages than in normal messages. Suppose a specified list of words is available and that your data base of 5000 messages contains 1700 that are spam. Among the spam messages, 1343 contain words in the list. Of the 3300 normal messages, only 297 contain words in the list. Obtain the probability that a message is spam given that the message contains words in the list.