Negative Downsampling and Recalibration

A CTR model is trained on 2 B impressions/day with a 0.5 % click rate. To fit the training budget you keep all positives and 5 % of negatives.

1. How many rows per day does the training set contain after downsampling, and what is the apparent click rate in the training data?
2. The model outputs p' = 0.20 for an impression. What is the calibrated probability for the real traffic distribution? Show the formula.
3. The predictions feed a second-price auction bidder that multiplies predicted CTR by the advertiser's value-per-click. Describe the concrete failure if the recalibration step is skipped.