Perform a cost$/$benefit analysis using fault$-$tree analysis on a stairway. Assume that over the last year, three accidents were caused by slippery surfaces, five were caused by inadequate railings, and three by someone negligently leaving tools or other obstacles on the steps. The average cost for each accident was $$200($total for first aid, lost time, etc.$).$ Assume that you have been allocated $$1,000$ to improve the safety of the stairway $($although you don$$t need to spend all of the money$).$ There are three alternatives which could be employed:

$1)$ new surfaces, which will reduce accidents caused by slippery surfaces by $70\%$ and cost $$800,$

$2)$ new railings, which will reduce accidents caused by inadequate railings by only $50\%($since not all pedestrians use railings!$)$ and cost $$1,000$

$3)$ signs and educational programs, which are estimated to reduce both railing and obstacle related accidents, each by $20\%($people forget easily$)$ but cost only $$100.$

$($To calculate basic event probabilities, assume the stairway is $5$ times per hour, $8$ hours a day, $5$ days a week, $50$ weeks a year.$)$

a$)$ Draw a fault tree of the situation. $($please do it on paper$)$

b$)$ Evaluate all alternatives $($or combinations$)$ to determine the best allocation of the $$1,000.$