Question

Software to detect fraud in consumer phone cards tracks the number of metropolitan areas where calls originate each day. It is found that 2% of the legitimate users originate calls from two or more metropolitan areas in a single day. However, 30% of fraudulent users originate calls from two or more metropolitan areas in a single day. The proportion of fraudulent users is 0.0164%. If the same user originates calls from two or more metropolitan areas in a single day, what is the probability that the user is fraudulent? Round your answer to six decimal places (e.g. 98.765432).

Answer 1

Let us first define the events

A event will be legitimate user

event will be fraudulent user

B event will be user originates calls from two or more metropolitan areas in single day

It is also given proportion of fraudulent users is 0.0164% so probability of fraudulent users will be

probability of legitimate users will be

It has also been mentioned in the  question that 2% of the legitimate users originate calls from two or more metropolitan areas in a single day

Here we will find probability

And 30% of fraudulent users originate calls from two or more metropolitan areas in a single day

Now we will find probability P(B)

after the above calculations

we have to calculate the probability that a user fraudulent given that the user originates calls from two or more metropolitan areas in a single day

Hence the required probability is 0.002454 approximately