Bayes' Rule

Question

The IRS checks the deduction for contributions to identify fraudulent tax returns. They believe that if a taxpayer claims more than a certain standard amount, there is a 0.20 probability that the return is fraudulent. If the deduction does not exceed the IRS standard, the probability of a fraudulent return reduces to 0.03. Suppose about 14% of the returns exceed the IRS standard.

(a) Estimate the percentage of returns that are fraudulent. (Enter your answer to two decimal places.)
%

(b) A concerned citizen informs the IRS that a certain return is fraudulent. Find the probability that its deductions exceed the IRS standard. (Round your answer to three decimal places.)

Answer 1

(a)

 

We'd like to find out the probability that a standard-exceeding return is fraudulent and the probability that a normal return is fraudulent, then add those together.

 

14% of the returns exceed the standard, and 20% of those are fraudulent. That's a probability of 0.14(0.2) = 0.028.

 

100 - 14 = 86% of the returns are normal, and 3% of those are fraudulent. That's a probability of 0.86(0.03) = 0.0258.

 

That's a total of 0.028 + 0.0258 = 0.0538 fraudulent returns, or 5.38%.

 

 

 

(b)

 

We'd like to find P(standard-exceeding|fraudulent), or P(E|F).

 

P(E|F) = (P(F|E)P(E)) / P(F)

 

 

P(F|E), or P(fraudulent|standard-exceeding), was given as 0.2.

 

P(E), or P(standard-exceeding), was given as 0.14.

 

P(F), or P(fraudulent), was just calculated as 0.0538.

 

 

So:

 

P(E|F) = (P(F|E)P(E)) / P(F)

P(E|F) = ((0.2)(0.14)) / 0.0538

P(E|F) = 0.028/0.0538

P(E|F) = 0.520

 

 

That's pretty crazy. Even if the IRS has methods in place to check the returns that are standard-exceeding, and only a tiny portion of regular returns are fraudulent, about 50% of fraudulent returns won't get caught.

 

That's because there are so many more regular returns than standard-exceeding ones... The "tiny portion" of that large number winds up being about the same as the fraudulent 20% of standard-exceeding returns.