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Answer:
Probability false alarms will go down- Probability it will go down after new system installed+ Probability it will go down without new system installation. Here it is assumed that the director will surely install the new sensors if budget is approved. D" be probability false alarm goes down. DwN" be probability it goes down with new system. DwoN" It goes down without new system.
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DwN 1/3(since odds director believes is 1 to 2) DwoN-1/(odss not reduced is 3:1 so reverse of that) Since new sensor depends upon budget approval. P(New Sensor)-1/2-P(No new sensor)-0.5 Required probability P(New Sensor)DwNP( No new sensor)DwON -0.5(1/3)+0.5(1/4)0.2917 So answer is b) less than even better than 25%.
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