IS THE WINNER REALLY A WINNER?

In a class, there are 50 students. 80% of them are just average students. But, remaining 20% is highly intelligent. In a objective type examination, one student has scored very high marks. Is he really intelligent? Or an average student scored top marks out of sheer luck? Let us find out.

The probability of intelligent students in the class is 80% or 0.80 -P(I). The probability of the average students present in the class is 20% or 0.20 -P(A). The probability of clever students winning-getting top marks is P(W/I) =0.60 (already known). The chance for average students to win is P(W/A) =0.20 (already known). The chance for winner being intelligent is got by applying Bayes theorem

P(I/W) = P(clever/win) = P(A)*P(W/I)/(P(A)*p(W/I)+P(I)*p(W/A))
= 0.20*0.6/(0.2*0.6+0.8*0.2)
=43%

43% is very less. We cannot conclude that the winner is really smart. Suppose, he scores well in the second test. He may be smart. The chance for smart boy to win second time is 0.6*0.6 =0.36 - P(W/I). The chance for average students to score high for the second time is 0.2*0.2 =.04 -P(W/A)- (It is all based on multiplication rule in statistics). Now, rewrite the Bayes theorem using the new probabilities.
P(smart/2 wins) = 0.2*0.36/(0.2*0.36+0.8*0.04)
=69%
Ok. The winner mostly may be clever. But still some doubt lingers. Why do not we go for a third test? The chance for smart student to win in all the three tests is 0.6*0.6*0.6 = 0.216. The chance for average student to win in three tests is 0.2*0.2*0.2 = .008. Now, using Bayes theorem, the three time winner being intelligent is 87%. Now we are satisfied.
It can be proved
P(smart/4 wins) = 95%
P(intelligent/5 wins) =98%

Hence, if a student scores high marks in five consecutive tests, he is really smart without any reasonable doubt. These results holds good, when the average and above average students are in the ratio 2:8. We have to recalculate, if the ratio and probabilities changes. Generally, if there are large number of low IQ students and a few high IQ students, we have to go for more tests.
This principle can also be applied to sports, games and selection process. We understand that consistency is important to win.
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