1. A software company is trying to use Bayes Theorem and otherrules of probability to develop an algorithm that can effectivelyfilter out spam emails. The company’s software developers carefullyexamined a random sample of 20,000 emails received by its employeesand found out that 4,400 of the emails were spam. A closer look atthe spam emails revealed that 1,100 of them contained the word‘free’ in the subject line. On the other hand, only 780 of thenon-spam emails had the word ‘free’ in their subject line. Pleaseanswer the following questions based on the given information.

a. What is the probability that a random email will be a spamand will not contain the word ‘free’ in itssubject line? Please show your work.           

b. If the word ‘free’ does not appear in thesubject line of an email, what is the probability that the email isnot a spam? Please show work.                 [2points]

1. c. What is the probability that a random email will beneither a spam and nor willcontain the word ‘free’ in its subject line? Please show yourwork.         [2points]                               
   1. d. Are an email being a spam and its subject line containingthe word ‘free’ independent events? Please showhow you arrived at your answer.     [1 point]