Negative Downsampling and Re-calibration

You train a pCTR model on 20 B impressions/day with a base CTR of 0.8 %. To cut cost you keep every click and 5 % of the non-clicks.

1. What is the positive rate in the training set after downsampling?
2. The model outputs raw probability q = 0.14 for an impression. What is the corrected pCTR that should be sent to the auction?
3. A colleague argues: "We use log loss, so the model is calibrated by construction — no correction needed." Explain what goes wrong if you skip the correction, and name an alternative that avoids a separate correction step.