The lie detector

At a store, one employee in ten is stealing merchandise.

The managers want to fire the thieves, but their only tool is a lie detector that is 80% accurate:

A thief fails the test 80% of the time.
An innocent employee passes the test 80% of the time.

An employee fails the test. What is the probability that this employee is a thief?

No employee picked yet

Pick an employee at random and give them the test.

Your tally

Rounds: 0
Failed the test: 0
Failed and a thief: 0

Test

?

Test result

?

The truth

00

No employees tested yet. Choose how many to test.

Estimate after each failed test

The fraction of employees who failed the test so far who are thieves.

Every employee tested so far

Group by outcome

Failed the test, thief 0
Failed the test, innocent 0
Passed the test, so the employee doesn’t count 0

Reveal the answer

The tempting answer is 0.8, because the test is 80% accurate. That answer ignores the base rate: only one employee in ten is a thief, so most of the employees who fail the test are innocent people caught by the test’s mistakes.

Picture 100 employees. The top row holds the 10 thieves, and the other nine rows hold the 90 innocent employees. The highlighted employees are the ones who fail the test.

The test catches 8 of the 10 thieves. It also wrongly fails 2 in every 10 innocent employees, which is 18 of the 90. So 26 employees fail the test, and only 8 of them are thieves. The probability that an employee who fails the test is a thief is 8/26 = 4/13, or about 0.31.

In symbols, let T be the event that the employee is a thief and F the event that the employee fails the test. By Bayes’ rule,

P(T | F) = P(F | T) P(T)P(F) = 0.8 × 0.10.8 × 0.1 + 0.2 × 0.9 = 0.080.26 ≈ 0.31.

A classroom demonstration by Frank DiTraglia.