What Is the Probability an Email Quarantined as Spam Is Actually Spam?

[gtfitzpatrick]

Homework Statement

45% of emails sent to my account is spam. i set up a filter but it fails toquarantine 2% of spam and in advertently quarantines5% of genuine emails.
What is the probability that an email will be quarantined as spam?
if an email is qurantined as spam what is the probability that is actually spam?

Homework Equations

The Attempt at a Solution

so i take it that this isconditional probabability so the I am looking at P(alb) = P(a$\cap$b) / P(b)

but I am not sure as to what to use as a and b
B is going to be the probability of it being spam in the first place which will be .45
as it misses 2% of spam and included 5% of genuine emails.
so is a 0.45 - 0.009 + 0.011 ?

 

[tiny-tim]

hi gtfitzpatrick!

to make it clearer, so you can use P(alb) = P(a$\cap$b) / P(b), rewrite …

What is the probability that an email will be quarantined as spam?
if an email is qurantined as spam what is the probability that is actually spam?

as …

If email what-proportion quarantined email ?

If quarantined email what-proportion spam quarantined email ?

 

[gtfitzpatrick]

hi Tim :)
if email-44.65 quarantined spam?(misses 2% of 45% + 5% of 55%?)

 

[lanedance]

To add on to Tiny Tim's hints, I would start by first defining a useful notation, how about the following:
Q - quarantined
nQ = q - not quarantined
S - Spam
nS = s - not Spam

Note that Q & nQ=q, and S & nS=s each are a closed set of multually exclusive events, that is
P(Q∩q) = 0
P(Q) + P(q) = 1
P(S∩s) = 0
P(S) + P(s) = 1

Now using the above terminoology, how would you write the following 3 phrases from the question:
probability of emails sent to my account is spam = 45%
probability of fails to quarantine of spam = probability of fails to quarantine given spam = 2%
probability of quarantines genuine email = probability of quarantines given genuine = 5%

Once you have all these it should be clearer where to go