Business Context

On Facebook, Integrity teams monitor outbound friend-request abuse because spammy requests can hurt the AARRR activation experience and downstream retention. Suppose Meta estimates that only a small share of accounts are bad, but those accounts send requests much more aggressively.

Problem Statement

A user on Facebook receives exactly one friend request during a short observation window. What is the probability that this request came from a bad account?

Given Data

Interpret the rate as a weighting factor: in the same time window, a bad account is 10 times as likely as a good account to generate a friend request event.

Requirements

1. Define the relevant events and write the conditional probability to compute.
2. Use Bayes' theorem to derive the probability that an observed request came from a bad account.
3. Show the equivalent solution using event-rate weighting rather than formal Bayes notation.
4. Compute the expected composition of requests if Meta observed 1,000,000 accounts in the same window.
5. Briefly explain why the answer is much larger than the 1% base rate.

Assumptions

• Each account is either bad or good.
• The 10x multiplier applies uniformly across the observation window.
• We condition on observing a request event, not on sampling an account uniformly.
• Ignore downstream ranking, enforcement, and FB Groups or Instagram surfaces; this is only about friend-request generation.