Question

Taxes are audited by automation initially and is prone to mistakes. The automated audit marks if an individuals’ taxes are completed correctly (by marking them with a G) or incorrectly (by marking with a B). Suppose that 3% of a populations’ taxpayers submit their taxes incorrectly. The automated audit, prone to mistakes, has a false positive rate (the probability of marking an individuals’ taxes with a B given the taxpayer did complete them correctly) of 6% and a false negative rate (the probability of marking an individuals’ taxes with a G given the taxpayer did not complete them correctly) of 8%. A taxpayer selected from this population was auto audited and found to have a positive result (marked with a B for did not complete correctly). What is the probability that the taxpayer did completed their taxes correctly? Apply Bayes' Rule to obtain your answer. (Round to four decimal places as needed.) *Start by defining your main events and conditional events you have info on*

Answer 1

Let T and ~T be the event that taxpayers submit their taxes correctly and incorrectly respectively.

Given,

P(~T) = 0.03

P(T) = 1- P(~T) = 1- 0.03 = 0.97

P(B | T) = 0.06

P(G | ~T) = 0.08

P(B | ~T) =1 -P(G | ~T) = 1 -   0.08 = 0.92

By law of total probability,

P(B) = P(T) P(B | T) + P(~T) P(B | ~T)

= 0.97 * 0.06 + 0.03 * 0.92 = 0.0858

Given that taxpayers was marked B probability that the taxpayer did completed their taxes correctly

= P(T | B)

= P(B | T) P(T)/P(B) (Bayes Theorem)

= 0.06 * 0.97 / 0.0858

= 0.6783