FALSE POSITIVE AND DISEASE

In tossing coin, each event (toss) is independent event. The outcome of one toss does not influence the next toss. Hence the chance of getting a head is always 1/2.
Suppose, you left your car key in one of the four rooms of your apartment. Let us name the rooms as A,B,C and D. The probability of finding the key in the room A is 1/4. That is written as P(A) =1/4.
You searched the room A thoroughly and failed to find the key. Now the probability of finding the key in the room B is 1/3. It is written as P(B/A)=1/3. It is read as "probability of B given A". Remember the symbol.

A new disease has stroked the people. But only one (1%) of the people is affected by that disease. A Scientific lab has made a new 'device and kit' which detects the disease with 98% accuracy. But there is a catch. If a person does not have the disease, the device will falsely recognize the disease with 2% chance. It is a 'false positive'. 98% accuracy seems to be good for us. But let us see, how the story goes

P(A) - probability of population having disease = 1% = 0.01
P(B)- probability of population having No disease = 99% = 0.99
P(C/A) - probability of disease detected in affected population = 98%=0.98
P(C/B) - probability of disease detected (falsely) in normal population 2%=.02
we want:
P(A/C) = probability of affected population really detected by the device.

In statistics, Bayes theorem states that
P(A/C) = P(C/A)*P(A)/(P(C/A)*P(A) +P(C/B)*P(B))
P(A/C) = 0.98*0.01/(0.98*0.01+0.02*0.99)=33.1%

Hence the device's accuracy has gone down from 98% to 33%. What happened? The disease is so rare. If you test the general public , the number of people not having the disease will greatly out number the small group that has it. So even a very low rate of false positives will strongly impact the overall results.
Bayes' theorem is very intriguing statistical tool. It has many interesting and important applications.
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