Question

1. A train station has installed a system for determining whether bags contain explosives. It has a 95% chance of correctly identifying a bag containing explosives (5% chance of a false negative), and a 99.5% chance of correctly classifying a bag without explosives as safe (0.5% chance of a false positive). Suppose that the train station screens 4 million bags per year, and that 10 of these bags are expected to contain explosives. (2.5x10^-6 probability that a bag contains explosives)
   1. A bag is identified by the system as containing explosives. What’s the probability that it actually contains explosives?
   2. If we want the probability in part (a) to be at least 0.5, what should the probability of correctly identifying a bag without explosives be?
   3. Would it be possible to make the probability in part (a) at least 0.5 by increasing the chance of correctly identifying bags containing explosives? Justify your answer

The probability of identifying a bag with explosives correctly is 95%
the probability of identifying a bag without explosives as safe is 99.5%
Given a sample space of 4 million, there are 10 bags that are expected to contain explosives
This is all the info the problem gives me

Answer 1

a)P(identified as containing explosives)=P(actually contains explosives and identified as containing explosives)+P(actually not contains explosives and identified as containing explosives)

=(10/(4*10⁶))*0.95+(1-10/(4*10⁶))*0.005 =0.005002363

hence  probability that it actually contains explosives given identified as containing explosives)

=(10/(4*10⁶))*0.95/0.005002363=0.000475

b)

let probability of correctly identifying a bag without explosives be a

hence a =0.99999763 ~ 99.999763%

c)

No as even if that becomes 1 ; proportion of true explosives will always be less than half of total explosives detected,