Business Context

You’re a data scientist at PayWave, a fintech processing 8–12 million card-not-present transactions per day across North America. Fraud losses are tightly monitored: every additional $10k/day in fraud is escalated, but overly aggressive blocking also harms customer trust and increases support costs.

A new real-time fraud model produces a binary signal Y = 1 (“alert”) when it believes a transaction is suspicious. The fraud operations team wants a simple, interpretable number: given an alert, what is the probability the transaction is actually fraudulent? This is crucial for deciding whether to auto-decline the transaction, step-up authenticate, or send to manual review.

Problem Statement

Compute the Bayesian probability P(X = 1 \mid Y = 1) where:

• X = 1 means the transaction is truly fraudulent.
• Y = 1 means the model raised an alert.

Then quantify what this implies operationally at PayWave’s scale.

Given Data

Assume the following metrics were estimated from a recent backtest on a representative week of traffic:

Requirements

1. Write Bayes’ theorem for $P(X=1 \mid Y=1)$ in terms of the given quantities.
2. Compute $P(Y=1)$, the overall probability a transaction triggers an alert.
3. Compute $P(X=1 \mid Y=1)$ numerically (this is the posterior fraud probability given an alert).
4. Convert the result into expected daily counts: expected number of alerts per day and expected number of true frauds among those alerts.
5. Briefly interpret: if PayWave auto-declines all alerts, what’s the precision implication and what trade-off does the base rate create?

Assumptions and Constraints

• Treat the backtest rates as stable and applicable to production (no distribution shift).
• Alerts are conditionally independent given fraud status (i.e., the model’s error rates summarize behavior).
• Ignore downstream effects (fraudsters adapting, customer churn due to false declines) for the calculation; discuss them qualitatively in interpretation.