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.
-----------------------------------------------------------------------------------------------

Comments

THE EARTH, A SUPER ORGANISM

JOIN MY COURSE: "Become a programmer in a day with python" A man called 'love lock' (what a name) proposed a theory called Gaia theory, named after Greek Goddess. It says, "Earth is a self-regulating organism like a human being. The organic life in it interacts with in-organic matter and maintains atmosphere, temperature and environment". Hence the earth is still suitable for the life to thrive. Imagine, in a particular place, there are lot of flowers. Some flowers are white and some are darkly coloured. We know, white reflects light and heat while dark absorbs the same. White flowers can thrive in hot climate. But dark flowers requires cold climate. The absorption and reflection balances and the environment reaches average, warm temperature at which both the flowers can co-exist. This is the essence of "Gaia" theory. On our earth, ...

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. ...

DISORDER IS THE "ORDER OF THE DAY"

Imagine a balloon full of air. The air molecules are moving randomly inside the balloon. Let us pierce the balloon with a pin. The air rushes out. Why should not the air molecules stay inside the balloon safely and ignore the little hole? That is not the way the world works. The molecules always "want to occupy as many states as possible". Hence the air goes out in the open to occupy more volume. The things always goes into disorder (entropy) and the disorder increases with time. The above statement is what we call "second law of thermodynamics". Consider a cup of coffee on the table. Suppose the heat from entire room flows to your cup of coffee, the coffee will boil and the rest of the room will freeze. Freezing means bringing things to order and arrangement. It violates the second law. Hence it will never happen. Hence heat must flow from high ...

sciencecapsules

Search This Blog