by using python

The five most common words appearing in spam emails are shipping!, today!, here!, available, and fingertips! (Andy Greenberg, The Most Common Words In Spam Email, Forbes website, March 17, 2010). 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!

.051

here!	.034
available	.014
fingertips!

.014

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

Word	Proportions of ham that have this word
shipping!	.0015
today!	.0022
here!	.0022
available	.0041
fingertips!	.0011 quetion

.014

quetions

a. What is the probability that an email includes the word "shipping!"?

b. 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?

# c. What is the probability that an email includes the word "today!"? # If a message includes the word today!, what is the probability the message is spam?

# d. If an email containing "shipping!" is independent of this emal containing "today", # what is the probability that an email contains both "shipping!" and "today"?