Your e-mail provider has a very good spam filter that only allows 1.5% of the to

Question

Your e-mail provider has a very good spam filter that only allows 1.5% of the total spam to not be filtered out. A lot of spam e-mails are trying to phish for your personal information so they ask you to go to an unknown web site where you are asked to enter this information. About 98% of spam will ask you to click on a link without providing you with the explicit url. In addition, 3% of legitimate e-mails will ask you to click on a link without the explicit url being shown. For a random e-mail in your inbox, if you are provided a link without the explicit url, what is the probability that the e-mail will be spam? BONUS: Why is your answer counterintuitive?

Explanation / Answer

Given information,

P(spam) = 0.015

P(not spam) = 1 - P(spam) = 1 - 0.015 = 0.985

and

P(without url | spam) = 0.98

P(without url |not spam) = 0.03

By the law of total probability, the probability that email provide a link withut the explicit url is

P(without url) = P(without url | spam)P(spam) + P(without url | not spam) P(not spam)= 0.98 *0.015 + 0.03* 0.985 = 0.0147 + 0.02955 = 0.04425

So the probability that email will be soam given that you are provided a link without the explicit url is

P(spam |without url) = [ P(without url | spam)P(spam)] / P(without url) = 0.0147 / 0.04425 = 0.3322

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Here we thought that spam mail should have greater chances to provided a link without explicit url. But spam mails are mostly filltered so there are greater chances that normal mail provided a link without explicit url so answer is counterintutive.

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