# Conditional probability

1. Mar 14, 2012

### gtfitzpatrick

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

3. The attempt at a solution

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

but im 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 ?

2. Mar 14, 2012

### tiny-tim

hi gtfitzpatrick!

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

If email what-proportion quarantined email ?

If quarantined email what-proportion spam quarantined email ?

3. Mar 14, 2012

### gtfitzpatrick

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

4. Mar 14, 2012

### 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:
Probabilty of emails sent to my account is spam = 45%
Probabilty of fails to quarantine of spam = Probabilty of fails to quarantine given spam = 2%
Probabilty of quarantines genuine email = Probabilty of quarantines given genuine = 5%

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