Consider the exams/competitions/selections or anything similar, where the final success/failure is decided in steps/stages. So, you clear the first stage, then go to the second and so on. Now, the peculiar thing is that every time you clear a stage, your hope (and more than that others’ hope for you) to be able to finally win starts getting stronger. Why should that happen? After all it is not at all difficult to see that competition becomes tougher in each stage. You stand lesser and lesser chance of *getting to the next stage* with each passing stage. Why still these escalating hopes? Am I missing something? Or is it just another one of the irrationalities of human minds?

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Since I am removing the Haloscan Comments, I am copy-pasting the comments I got on this post here.

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Jaya:

Why should that happen?

Because sky have no limits and we have got wings as well energy to fly.

Why still these escalating hopes?

Because we want to catch the sun.

Am I missing something?

Yes,A bit of confindence!

is it just another one of the irrationalities of human minds?

No,this is a rational thinking when we want Miles to Go…..

Prem Piyush | Email | Homepage | 12.05.04 – 8:37 pm | #

I am having readers who are talking on a higher plane than I am!!

When did I say that any thing was wrong with having hopes. In fact that is not the point of talk at all.

I was rather talking of those phenomena, “Oh! You got through the prelims; since you have reached upto here, I am sure you will make it.”

There is absolutely no reason to think that “since one has reahced upto step n, it is easy to reach upto step n+1”. Confidence, hopes (as such), skies are out of the domain of this post of mine.

Jaya Jha | Homepage | 12.05.04 – 10:00 pm | #

Gravatar There are very few persons who make me feel in which plane I stand.

May some day I reach upto plane of readers.

Prem Piyush | Email | Homepage | 12.06.04 – 11:46 am | #

Gravatar Let�s assume 5 contestants & 4 successive stages for selection of a winner. For a rational process to select a winner there must be some difference in abilities of the contestants.Let the ability of C1=1;Ability of C2=2 and so on.If the contest is fair, C5 must be the winner.Let�s find the probability of C5�s win at each stage: At stage 1 P( C5’s win) =5/15=0.333; At stage 2, P=5/14=0.357 & so on and at stage 4, P=5/9=0.555. Though I�ve taken specific nos here, a general proof is easily designed. The interesting observation is that C5�s probability of winning after each stage increases at an increasing rate.The above calculations are based on the assumptions of a fair process, but if the process is unfair (a person with lesser ability sneaks thru) then the specific parameters of the unfairness would be required for calculations.Are we in a society where selections are fair…that�s a value judgment currently beyond mathematics (Did I miss out something?)

Ayan Bhattacharya | Email | 12.06.04 – 3:45 pm | #

First of all this simple number based probability would not work. Probability of success does not just depend on the number of competitors, but also their abilities. Otherwise recruiters could have as well picked up people randomly, rather than designing elaborate procedures (one might believe that it would be pretty good a system , but point is that that’s not a system. And some ability, whether or not the right one, is looked for in the process.)

But if we assume to the process to be fair, there is no question of probability and hence expectation at all. It should be obvious that the person with best abilities would be selected.

Jaya Jha | Homepage | 12.07.04 – 1:15 pm | #

Reason everyone still appears for the exam/competition is that one may not know whether or not one has the best ability. So, one would hope and write the exam anyway. But even this can not explain escalating hopes or any particular degree of hope, since one does not know the ability of others in comparison to oneself.

So, the presence of hope is there because of probabilistic nature of the process. i.e. the selection process is “not-so-fair” after all. There is a finite probability that even if one has an ability of 4, and the other an ability of 5, there is some finite chance of the former getting selected and not later. Of course, it should not be a bad assumption that higher your ability is higher your chances of getting through, and higher the ability of your competitor is, lesser you chances of succeeding. Though even for the person of highest ability the chance of success is not 1.

Jaya Jha | Homepage | 12.07.04 – 1:16 pm | #

In this model let’s look at the scene. In the initial stages there would be many people with low abilities. By the above hypothesis, the number of people with lower abilities would become lesser and lesser in the successive stages (since their probability of succeeding is low). If say 50% people are selected at each stage, it is easier to be in top 50% in a large population where there would be many people with low abilities than in a smaller population where others, on an average, would have a considerable higher ability. So, with each successive stage the chances should be less.

I do not have much fancy for numbers; someone interested may try to see, if this argument fits mathematics

Jaya Jha | Homepage | 12.07.04 – 1:16 pm | #

Gravatar If I�ve understood right, the following problem summarizes your misgivings:

Problem: Consider a population of 100; 80 people with ability 1, 19 with ability 2 and 1 with ability 3. The aim is to select one winner, through successive selection stages, each stage eliminates 50% of the population.

Soln: At Stage1, Probability of a person with ability 1 getting selected=3/8=0.375

After the elimination at Stage 1, we have a total population of 50 with 30 persons with ability 1.

At Stage 2, Probability of a person with ability 1 getting selected=5/30=0.1666

After the elimination at Stage 2, we have a population of 25 with 5 persons with ability 1.

At Stage 3, Probability of a person with ability 1 getting selected=0.

Probability of a person with ability 2 getting selected=11/19=0.579

After elimination at Stage 3, we have a population of 12 (a little approx here since the no 25 is odd), with 11 persons having ability 2.

Ayan Bhattacharya | Email | 12.07.04 – 5:45 pm | #

Gravatar contd…

At Stage 4, Probability of a person with ability 2 getting selected=5/11=0.4545

Similarly, At Stage 5, Probability of a person with ability 2 getting selected=2/5=0.4

Finally, At Stage 6, Probability of a person with ability 2 getting selected=0, and we have a winner.

Your concern is that, say for a person with ability 1, from the nth Stage to the (n+1)th stage, the probability of selection is decreasing (and they think otherwise). But isn�t this decrease obvious (and people do realize this)? In this case the 50% elimination rule is not the same for all people (with different abilities), in Stage 1, it translates to a 62.5%(50/80) probability of elimination for people with ability 1, and 0% prob. of elimination for people with abilities 2 and 3. Similar translation of the general elimination rule for people with different abilities continues for the other stages.

Ayan Bhattacharya | Email | 12.07.04 – 5:48 pm | #

Gravatar contd….

An intuitive method of conceiving this problem is to imagine a pile of bricks one on top of the other. The length of each higher brick is half of the length of the brick below it. The bricks are arranged so that they are left aligned.

Now I make 100 equal notches on the lowest (the longest) brick from right to left, with the first 80 notches from right representing people with ability 1, next 19 people with ability 2 and the last notch, the person with ability 3.

At every stage of elimination, I�m breaking off the exposed portion of a brick starting from the brick at the lowest level. Thus the probability of selection of the person with ability 3 is 100% throughout, since his notch comes last.

Many interesting variations of this intuitive model (it sure isn�t intuitive when I�m trying to put it in words ) can be created.

Ayan Bhattacharya | Email | 12.07.04 – 5:50 pm | #

Gravatar the last in this series …

Finally, to answer your original misgiving (escalating hopes), the moment I know there is (a group of people better than me in the process) > (number to be selected), I know I�m out of luck. I think every rational human being realizes this; nobody expects to become the Formula 1 champion beating Schumacher (whatever the no of stages and the elimination percentage) unless he�s better than Schumacher….will that ever happen till Schumacher gives up voluntarily …..

Ayan Bhattacharya | Email | 12.07.04 – 5:52 pm | #

Ayan I think u nead to hear this:

“Because it is the people who are crazy enough to think they can change the world are the one who actually do.”

And the reason for that is there is no seperate measurement of abilities at hand other than the success on some test. What proof of being the best does Schumi has apart from being a constant winner?

Also U conviniently assumed the abilities and computed the probablities but a person inside the system cannot compute those probablities cause he has no way of knowing these abilities before hand.

And the same argument can also explain the escalating hopes also because with every step u are more sure of ur abilities. Agree that others are also stronger now but u are also in the same league. In other words, the difference between the top abilitiy and ur own perceived ability decreases.

whatever

abhaya | 12.08.04 – 3:09 pm | #