Answer to Question #347000 in Statistics and Probability for Nzuzi

Question

Question 1.12 [2, 3]

Catherine has a Gmail account and categorises her emails according to work and non-work related

emails. The probability that an email is a work-related email is 65.32%. Suppose furthermore it is

given that the probability that a work-related email received is a spam email is 15.34% and that if it

is a non-work related email that it is spam is 5.6%.

Calculate the following probabilities

a) That an email received by Catherine is a spam email.

b) Given that the email is spam what is the probability that it is a non-work-related email.

Answer 1

Solution:

Let's denote given values:

$P(w)=0.6532$ - probability of work-related emails;

$P(nw)=1-P(w)=0.3468$ - probability of non-work-related emails;

$P(ws)=0.1534$ - probability of work-related spam emails;

$P(nws)=0.056$ - probability of non-work-related spam emails.

Find:

a) $P(s)$ - probability of spam emails;

b) $P(s1)$ - probability of non-work-related spam emails, when received email is spam.

We will use general formula of calculation probability:

a) $P(s)=\frac{Ns}{Nw+Nnw}$ , (1)

$Ns$ - number of spam emails;

$Nw$ - number of work-related emails;

$Nnw$ - number of non-work-related emails;

We know:

$Nnws=NnwP(nws)$ - number of non-work-related spam emails;

$Nws=NwP(ws)$- number of work-related spam emails;

$Ns=Nws+Nnws;$

$\frac{Nw}{Nnw}=\frac{P(w)}{P(nw};$

So, if we put all of them to (1) formula:

$P(s)=\frac{P(nw)P(nws)+P(w)P(ws)}{P(w)+P(nw)}=\frac{0.3468\times0.056+0.6532\times0.1534}{0.6532+0.3468}=0.1196;$ or $P(s)=11.96$ %.

b) $P(s1)=\frac{Nnws}{Ns}=\frac{Nnws}{Nnws+Nws};$

$Nws=Nnws\times\frac{P(w)P(ws)}{P(nw)P(nws)};$ So,

$P(s1)=\frac{P(nw)P(nws)}{P(nw)P(nws)+P(w)P(ws)}=\frac{0.3468\times0.056}{0.3468\times0.056+0.6532\times0.1534}=0.1623;$

or $P(s1)=16.23$ %.

Answer:

a) $P(s)=11.96$ %;

b) $P(s1)=16.23$ %.