The Detector That Never Looks

A card issuer's fraud detector reports one number per transaction: the probability that the transaction is fraudulent. The other option, legitimate, gets whatever is left.

Its training set holds 5,000 transactions, of which 100 are fraudulent.

Before training anything real, the team builds a baseline that never looks at the transaction. It outputs the same probability $q$ for every transaction, and the team may pick any $q$ strictly between $0$ and $1$.

The loss is binary cross-entropy, averaged over all 5,000 transactions. For one transaction it is

$-\ln(\text{probability the detector gave to the true answer})$

so a fraudulent transaction costs $-\ln q$ and a legitimate one costs $-\ln(1 - q)$. Here $\ln$ is the natural logarithm.

With $q$ chosen as well as possible, what is the lowest average loss this baseline can reach?

Round to 3 decimal places.