Problem 1 (Base rate fallacy) A hypothetical country is looking to buy a new RADAR system to tighten it's border security. After a lot of market research, it has narrowed it's options down to one of 2 RADARs with the following specifications:

1. RADAR A: (a) Probability of issuing an alert given an enemy is present is 99% (b) Probability of issuing an alert given no enemy is present is 10% 2.

RADAR B: (a) Probability of issuing an alert given an enemy is present is 95% (b) Probability of issuing an alert given no enemy is present is 2%

After installation, the RADAR will be connected to an automated missile launcher which will fire every time the RADAR issues an alert. The cost of firing each missile, is sadly 1% of the yearly earnings of the country's population (Naturally, the country would want to minimize wastage while also wanting to maintain security).

Given that the probability of an enemy attacking the borders is 0.25, which RADAR should the country buy? Justify your choice numerically.