What is the probability that an e-mail containing the word 'Win' is a spam? A study reveal that 70% of all e-mails are spam. 90% of those contains the word 'Win', but 5% of regular e-mail also contain the word 'Win'. What is the probability that an e-mail containing the word 'Win' is a spam?
I know this is basic conditional probability but for some reason i cant get the right answer. Any help to point me in the right direction would help me alot.
Thank you
 A: Hint: Suppose you get $1000$ emails


*

*How many do you expect to be spam?

*How many do you expect to be not spam?

*How many of the spam emails do you expect to have "Win"? 

*How many of the not spam emails do you expect to have "Win"? 

*How many of the emails do you expect to have "Win"? 


Can you now find the probability that an email containing the word 'Win' is a spam?
A: Try dividing your work into cases:

  
*
  
*What is the probability that an actually spam email has 'Win' in it?
  
*What is the probability that a non-spam email has 'Win' in it?
  

How would you combine the previous two cases to get the total probability?
If you need more help, just ask me.
A: Remember Bayes' Theorem: P(a|b) = P(b|a)*P(a)/P(b).  
In this case, in pseudo-math, we want P(spam|contains word "win"). Plugging into Bayes' Theorem, we get that it should be:  
P(contains word "win"|spam)*P(spam)/P(contains word "win").  
Based on your numbers, P(spam) is 0.7 and P(contains word "win"|spam) is 0.9. Now, what's P(contains word "win")? The easy answer is 0.05 -- after all, that's the number the problem gives, right? Well, not quite. That 0.05 is only the REGULAR emails that contain the word "win" -- that is, it's showing P(contains word "win"|not spam). You need the "true" P(contains word "win") -- the probability that an email contains the word win regardless of whether it's spam or not.
