Solution
The Ledger That Loves Fives
Show the problem again
Honestly recorded amounts have a uniformly random final digit (0 through 9, independently across records). If a ledger has been fabricated, its author favours round numbers, and each record's final digit is 0 or 5 with probability 1/2 each, independently across records. A ledger is fabricated with prior probability $1/5$. An auditor samples two records from one ledger and finds that both end in 0 or 5. What is the posterior probability that the ledger is fabricated? Give a reduced fraction.
Worked solution
Likelihood of the evidence "both final digits lie in {0, 5}": for a fabricated ledger it is 1; for an honest one it is $(2/10)^2 = 1/25$. Posterior odds $= \frac{1/5\cdot 1}{4/5\cdot 1/25} = \frac{25}{4}$, so the probability is $\frac{25}{29}$. The answer is **25/29**. One such record alone would give only 5/9: each additional record multiplies the odds by 5, because digit evidence from independent records compounds. A single anomaly is weak; a handful together are damning.
Source: Terminal-digit analysis used in forensic accounting and data-integrity checks (cf. Nigrini on digit tests). Statement written for AxiomIQ.