An organization uses a spam filtering software to block 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:

• 96% of the spam messages are blocked in the High-Security-Mode.
• 90% 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:

• 10% 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

The organization estimates that 80% of the messages that it receives are spam messages.

(a)Let denote the conditional probability that a message that is not blocked by the spam filter operating in the High-Security-Mode is actually a spam message. Estimate .

(b)Let denote the conditional probability that a message that is blocked by the spam filter operating in the High-Security-Mode is actually not a spam message. Estimate .

(c)Let denote the conditional probability that a message that is not blocked by the spam filter operating in the Low-Security-Mode is actually a spam message. Estimate .

(d)Let denote the conditional probability that a message that is blocked by the spam filter operating in the Low-Security-Mode is actually not a spam message. Estimate .

(e)If the cost of not blocking a spam message is $1 and the cost of blocking a non-spam message is $10, should the organization operate the spam filter in the High-Security-Mode? Why?

(f)Recall that the cost of not blocking a spam message is $1. At least how high should the cost of blocking a non-spam message be for a risk-neutral rational decision maker to prefer operating the spam filter in the Low-Security-Mode?

(g)Let be the amortized cost per message of operating the spam filter. Explain at most how high could be for the organization to use the spam filter with the specified performance?