As per the given question: B is taken as the event that message is blocked and S is taken as the event that message is a spam
P(S)= 0.80 ⇒P(S̅)= 0.20
For High Security Mode: For Low Security Mode:
P (B|S)= 0.90 P (B|S)= 0.10 P (B|S)= 0.10 P (B|S)= 0.90
P (B|S) = 0.04 P (B|S) = 0.96 P (B|S) = 0.96 P (B|S) = 0.04
a. PH (B and S) = P(B|S) * P(S) = 0.10 * 0.20 = 0.02
PH = P (B and S) = P (B|S)∗P(S)=0.04∗ 0.8=0.032
Expected cost in High Security Mode= 0.02 * 10 +0.032 * 1 = 0.232
PL P (B andS ) = P (B|S) * P (S) = 0.04 * 0.2 = 0.008
PL= P (B and S) = P (B|S)∗ P(S)= 0.10 * 0.8 = 0.08
Expected cost in Low Security Mode= 0.008 * 10 + 0.08 *1 = 0.16
It is recommended that the organization should operate the spam filter in low security mode, as it is cheaper to operate.
b. Let the cost of blocking a non-spam message
Expected cost in high security mode is:
$1 * 0.04 * 0.80 + C * 0.10 * 0.20 = 0.032 + 0.02 C
Expected cost in Low Security mode is:
$1* 0.10 * 0.80 + C * 0.04 * 0.20 = 0.08 + 0.008 C
By solving the above equations, we get the breakeven cost
0.032+0.02 C = 0.08+0.008 C
0.032-0.08 = (0.008 – 0.02) C
C = 0.048/0.012
In order to prefer operating the spam filter in the Low-Security-Mode, we have to find the minimum value of $C for which the expected cost of operating in Low-Security-Mode is lower than that of operating in High-Security-Mode. A risk-neutral rational decision maker to prefer operating the spam filter in the Low-Security-Mode when its expected cost is lower.
c. Without a spam filter in place we won’t be blocking any messages. Under our current assumptions, the resulting expected cost per message is $0.80 since 80% of the messages are spam and the cost for not blocking a spam is estimated to be a dollar. How much does the spam filter saves us per message?
In my opinion, the organization should prefer using a spam filter operated in Low-Security-Mode. We are aware that 80% percent of the messages received by the organization are spam. Under the current assumption, the cost of not blocking a spam message or not using a spam filter is $1 per message; due to this we incur a cost of 80 cents per message.
The expected cost of operating a spam filter in Low-Security-Mode is $0.16 per message which is less than the cost incurred by not using a spam filter. If we use spam filter in Low-Security-Mode, we save 64 cents per message which makes it a preferred operating mode of spam filter.
As, calculated earlier, it is best to operate the spam filter in low security mode and expected cost to operate is $0.16. Thus, the spam filter saves us $0.80 – $0.16 = $0.64.
Hence, the maximum we should be willing to invest in the spam filter per message is $0.64.
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