Probability using spam filters

[arithmetic_amy1011]

Can anyone help please? I'm not sure if i worked it out correctly and I need to make sure as it is part of my assignment. Thank you!
A spam filter is designed to identify commonly occurring phrases in spam. Assume that 80% of Mike’s email is spam and that in 10% of the spam emails, the phrase “free credit” is used. On the other hand this phrase is only found in 1% of non-spam emails. If Mike receives another email that includes the phrase “free credit”, what is the probability that it is spam?

 

[Dr.Peterson]

Have you learned Bayes' Theorem?

Alternatively, out of 1000 emails, how many will fall in each category: spam with phrase, spam w/o phrase, non-spam with phrase, non-spam w/o phrase? Then find the conditional probability.

 

[arithmetic_amy1011]

@Dr.Peterson I tried working it out using Bayes' Theorem and my answer resulted in to 28.78%. Is this correct?

 

[arithmetic_amy1011]

I think the first part of the question is correct. However, im not really sure what has to be done in the second part. The second part of the question is: ii. Mike installs a second spam filter. An email arrives which is either legitimate (event L) or spam (event L c ). Program i marks it as legitimate (event Fi) or spam (event F c i ). Assume that the outputs from the two spam filters are conditionally independent and that both spam filters are 90% accurate, meaning that P(Fi |L) = P(F c i |L c ) = 9 10 . Find the probability that an email that has just arrived is legitimate, given that both spam filters mark it as legitimate