An organization uses a spam filtering software to block email messages that may potentially be spam messages. The spam filter can be set to one of two security modes: High-Security-Mode or Low-Security-Mode.

Extensive evaluation using a benchmark corpus consisting exclusively of spam messages yields the following performance statistics for the spam filter: 95% of the spam messages are blocked in the High-Security-Mode. 92% of the spam messages are blocked in the Low-Security-Mode.

Extensive evaluation using a benchmark corpus consisting exclusively of non-spam (legitimate) messages yields the following performance statistics for the spam filter: 8% of the non-spam messages are blocked in the High-Security-Mode 4% of the non-spam messages are blocked in the Low-Security-Mode

Over an initial test period the spam filter is operated in the High-Security-Mode. In this mode 77.6% of all messages are blocked. Let denote the conditional probability that a message blocked by the spam filter operating in the High-Security-Mode is actually not a spam message. Estimate .

Over the next test period the spam filter is operated in the Low-Security-Mode. In this mode 72.8% of all messages are blocked. Let denote the conditional probability that a message blocked by the spam filter operating in the Low-Security-Mode is actually not a spam message. Estimate

There are costs associated with not blocking spam messages and blocking non-spam messages. Let   

At least how low should be for a risk-neutral rational decision maker to prefer operating the spam filter in the High-Security-Mode?

The organization estimates that the cost of blocking a non-spam (legitimate) message is $20 and the cost of not blocking a spam message is $1. Let be the (amortized) cost per message of operating the spam filter. At most how high can be for a risk-neutral rational decision maker to operate the spam filter?