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10% of all email you receive is spam. Your spam filter is 90% reliable: that is, 90% of the mails it marks as spam are indeed spam and 90% of spam mails are correctly labelled as spam. If you see a mail marked spam by your filter, what is the probability that it really is spam?

  1. 10%
  2. 50%
  3. 70%
  4. 90%


asked in Probability by Veteran (92.5k points) 971 2329 3115 | 226 views

1 Answer

+3 votes
Best answer
10% email are spam, i.e. 90% email are not spam

90% of mail marked as spam is spam, 10% mail marked as spam are not spam

By Bayes theorem the probability that a mail marked spam is really a spam  

$$\begin{align*} &=\frac{\text{Probability of being spam and being detected as spam}}{\text{Probability of being detected as spam}}\\    &=\frac{0.1 \times 0.9}{(0.1\times 0.9) +(0.9\times 0.1)}\\&= 50\%\end{align*}$$
answered by Veteran (65.1k points) 35 222 625
edited by

90% of the mails it marks as spam are indeed spam

Can we say 90% from this statement alone?

Yes I think so :)

50% should be the right answer.

Consider I received 100 emails,

90 of them are actually non-spam, 10 are actually spam. 

Spam filter is 90% reliable, 

Out of 90 emails that are actually non-spams, with its 90% reliability, it will mark 81 as "non-spam" and 9 as "spam".

Out of 10 emails that are actually spams, again with its 90% reliability, it will mark 9 as "spam" and 1 as "not- spam".

So overall out of 100 it will mark total 18 emails as spam out of which 9 are actually spams & rest 9 are false positives.

so 9/18 = 50%.

P(Being detected as spam) = P(Actually Spam) x P(Being detected as spam | Actually Spam) + P(Actually Non-Spam) x P(Being detected as spam | Actually Non Spam).

= 0.1 x 0.9 + 0.9 x 0.1 = 0.18 

yes, the second part of denominator was wrong. Corrected :)
@srestha may you please tell in the correct format of Bayes theorem i.e.

P(E/F) = $\frac{P(F/E) * P(E)}{ P(F/E) * P(E) + P(F/E') * P(E')}$

that will be realy helpful.

Thanks in advance.
Can u plz derive that formula u have written?

@Shubhanshu P(Its Spam / Marked as Spam) = $\frac{P(Marked as Spam/Its Spam)*P(Its Spam)}{P(Marked as Spam/Its Spam)*P(Its Spam)+P(Marked as Spam/Its Not Spam)*P(Its Not Spam)}$ =$\frac{0.9*0.1}{0.9*0.1 + 0.1*0.9}$

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