Answer
a)In the event that the proposal passes, A can be written in terms of \[{{A}_{i}}\]as:
\[A=({{A}_{1}}\cap {{A}_{2}}\cap {{A}_{3}})\]
b)In this scenario, A can be written in terms of \[{{A}_{i}}\]as:
\[A=({{A}_{1}}\cup {{A}_{2}}\cup {{A}_{3}})\]
Work Step by Step
(a)
Let \[{{A}_{i}}\]denote the event that vice president i favors the proposal, i = 1, 2, 3, and let A denote the event that the proposal passes.
In first protocol, the proposal is to pass when all three vice presidents concur.
Therefore, the event that the proposal passes, A, can be written in terms of \[{{A}_{i}}\]as:
\[A=({{A}_{1}}\cap {{A}_{2}}\cap {{A}_{3}})\]
This type of system would be preferable to the other where new guidelines are being framed and requires all unit heads to concur.
(b)
Let \[{{A}_{i}}\]denote the event that vice president i favors the proposal, i = 1, 2, 3, and let A denote the event that the proposal passes.
In second protocol, the proposal is to pass when at least one vice president has passed the proposal.
Therefore, the event that the proposal passes, A, can be written in terms of \[{{A}_{i}}\]as:
\[A=({{A}_{1}}\cup {{A}_{2}}\cup {{A}_{3}})\]
This type of system would be preferable in emergencies where all unit heads may or may not be available; in that case, one head can pass the proposal.
