Statistics for finance · level 3
Bayes on a 2×2 table
Never manipulate Bayes' formula in your head. Imagine 100 (or 1,000) cases, fill the four boxes, and read the answer as a share of the row you are told you are in.
Worked example: 10% of borrowers default. A screen flags 90% of defaulters and 20% of good borrowers. A borrower is flagged — chance they default?
true positives9
false positives18
P(default | flag)33.33%
Step by step
- 1
Work in 100 borrowers, not probabilities
10 defaulters, 90 good
- 2
True flags
10 × 90% = 9
- 3
False flags
90 × 20% = 18
- 4
Share of all flags that are real
9 / (9 + 18) = 33.33%
10% of borrowers default. A screen flags 90% of defaulters and 20% of good borrowers. A borrower is flagged — chance they default? = 33.33
The theory behind it
Intuition
On a 2×2 table, the posterior is just the true-positive cell divided by all positives. Rare events make even accurate tests mostly wrong.
Common pitfalls
- ×Ignoring the base rate.
- ×Confusing sensitivity with precision.
In the interview
Screening, fraud and default-rate rounds.