Question

Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day.1% of legitimate users originate calls from 2+ metropolitan areas in a single day. 30% of fraudulent users originate calls from 2+ metropolitan areas in a single day. The proportion of fraudulent users is 0.01%. If the same user originates calls from 2+ metropolitan areas in a single day, what is the probability that the user is fraudulent?

Answer 1

Hmm, I'm not seeing where some of those numbers came from. Here's how I see it (using 1,000,000 users to avoid decimals)

of 1,000,000 users, \(0.01\% = \frac{0.01}{100}*1000000 = 100\) are fraudulent. Therefore, \(1000000-100=999900\) are legitimate.

The number of legitimate users who originate calls from 2+ metropolitan areas in a day is \(1\% = \frac{1}{100}*999900 = 9999\)

The number of fraudulent users who originate calls from 2+ metropolitan areas in a day is \(30\% = \frac{30}{100}*100 = 30\)

\[P[\text{caller from 2+ areas is fraudulent}] = \frac{\text{fraudulent callers from 2+ areas}}{\text{all callers from 2+ areas}}\]\[ = \frac{30}{30+9999} \approx 0.00299\]