EX 2.30 spam email

• An individual has 3 different email accounts.

  • Most of her messages, in fact 70%, come into account #1,
  • whereas $20\%$ come into account #2
  • and the remaining $10\%$ into account #3.

• Of the messages into account #1, only 1% are spam,

  • whereas the corresponding percentages for accounts #2
  • and #3 are $2\%$ and $5\%$ , respectively.

• What is the probability that a randomly selected message is spam?

• To answer this question, let’s first establish some notation:

  • $A_i=\{$ message is from account $\#i\}$ for $i=1,2,3,$
  • $B=\{$ message is spam $\}$

• Then the given percentages imply that

  • $P\left(A_1\right)=.70$
  • $P\left(A_2\right)=.20$
  • $P\left(A_3\right)=.10$

• We have

  • $P\left(B\mid A_1\right)=.01$
  • $P\left(B\mid A_2\right)=.02$
  • $P\left(B\mid A_3\right)=.05$

• Now it is simply a matter of substituting into the equation for the law of total probability:

$$P\left(B\right)=\left(.01\right)\left(.70\right)+\left(.02\right)\left(.20\right)+\left(.05\right)\left(.10\right)=.016$$

• In the long run,
  • $1.6\%$ of this individual’s messages will be spam.