Question

7. When you submit your tax return to the IRS, they screen your return for indications of fraud. Naturally the screening system is not perfect: only 92% of fraudulent tax returns get red-flagged (that is, the IRS’ system detects the fraud), and 9% of honest tax returns get red-flagged (the system mistakenly believes these good returns to be fraudulent). A tax return the IRS believes not to be fraudulent is said to be “green-flagged.” Suppose that 4% of all tax returns are fraudulent. (a) What percent of all tax returns are correctly flagged?

(b) If a tax return is red-flagged, what is the probability this return is fraudulent?

(c) The answer to part (b) might worry an administrator at the IRS (or a taxpayer, for that matter!). Briefly explain what causes the answer to part (b) to be somewhat low.

Answer 1

We have given below information
p( flagged red / honest tax returns) = 0.09
p( flagged green / honest tax returns) = 0.91
p( flagged red / fraudulent tax returns) = 0.92
p( flagged green / fraudulent tax returns)= 0.08
p(honest tax returns) = 0.96
p(fraudulent tax returns)= 0.04

(a) What percent of all tax returns are correctly flagged?

We have to use Bayes' theorem to calculate conditional probabilty

We have to calculate

p( fraudulent tax returns,tax returns flagged red ) +p( honest tax returns,flagged green )

p( fraudulent tax returns,tax returns flagged red )=p( flagged red / fraudulent tax returns)*p(fraudulent tax returns)
=0.92*0.04
=0.0368
p( honest tax returns,flagged green )=p( flagged green / honest tax returns) *p(honest tax returns)
=0.91*0.96
=0.8736
=0.9104

(b) If a tax return is red-flagged, what is the probability this return is fraudulent?

p( fraudulent tax returns/flagged red)=p( fraudulent tax returns,tax returns flagged red )/p(flagged red)

p(flagged red)=

p( flagged red / fraudulent tax returns)*p(fraudulent tax returns)+p( flagged red / honest tax returns)*p(honest tax returns)
=0.92*0.04+0.09*0.96
=0.1232
p( fraudulent tax returns/flagged red) =0.0368/0.1232
=0.2987

(c) The answer to part (b) might worry an administrator at the IRS (or a taxpayer, for that matter!). Briefly explain what causes the answer to part (b) to be somewhat low.

the answer to part (b) to be somewhat low because p(fraudulent tax returns) is small as 4%

and p( flagged red / honest tax returns) is also small 9%