No, this is not a post about JEE. You can read about that here and here.
This is a post about a lesser known entrance exam - one conducted on the Planet Dork in the Galaxy of Styrofoam.
Before we get to the exam itself, you must understand a little bit about Dorkian society. The entire population of Dork is divided into two equal categories - X and Y. This division is not based on ability - people in both categories are equally intelligent - yet Dorkian society defines different roles for them.
Type X people are supposed to be primarily responsible for domestic arrangements, and for looking after young Dorkians. While there is a growing trend of type X folks seeking employment opportunities and career, Dorkian society does not take seriously the idea of Xs as primary wage earners. As a result, Dorkian caregivers place less emphasis on education for Xs - they believe the returns to such investments (both in general and more specifically, for themselves) will be relatively poor, and while they're happy to encourage Xs to pursue careers, they don't see it as high priority.
By constrast, type Y people have a clearly defined role as primary income earners, and having a successful career is seen as the highest priority for them. As a result, society is willing, even eager, to make whatever investments in their education are felt necessary.
The most preferred career track for potential employment seekers on the planet Dork is to become an Alien. Being an Alien means plenty of travel, lots of BPs (the Dorkian currency), maybe even the opportunity to star in a Spielberg film. Becoming an Alien is not easy though. Precisely because Alien status is so sought after, competition is fierce, and young Dorkians spend sun-cycles of their lives preparing to get into a good Alien training institute.
The most sought after of these are the Institutes for Inter Terrestrial Studies (the IITS). Every year, hundreds of thousands of Alien hopefuls sit for the IITS Entrance Test (ET), hoping to make it into those hallowed halls. IITS are not the only places that offer degrees in Alien-hood, though - there are hundreds of other, less prestigious institutes, collectively known as the Also There Institutes (ATIs).
Right. With that set-up, let's consider a set of young Dorkians who have come of age and are considering a future career (and for the moment, let's assume they're all interested in being Aliens). According to the Falstaffian Academy of Random Trivia - the premier statistical agency on the planet Dork - the probabilities around the ET are as follows:
Probability of young Dork having any potential to be an alien: X: 10%, Y: 10%
Probability of Dork with potential taking ET / other Alien exam: X - 50%, Y - 100%
Probability of ET candidate passing in first attempt: X - 1%, Y- 1%
Probability of failed candidate taking ET again (2nd attempt): X - 5%, Y - 75%
Probability of ET candidate passing in second attempt: X- 3%, Y - 3%
Probability of failed candidate taking ET again (3rd attempt): X - 0%, Y - 20%
Probability of ET candidate passing in third attempt: X - na, Y - 5%
Three things are important to note about these statistics.
First, that the probability of passing in the ET is independent of type. There are no systematic differences in ability / preparedness between the two types.
Second, that the societal bias against X types gets reflected in two ways - first, in a lower probability of their ever attempting to sit for the ET (50% vs. 100%). And second in a dramatically lower probability of their undertaking subsequent attempts after they fail the first time around. These lower probabilities reflect the lower priority Dorkian caregivers (and perhaps X types themselves) place on becoming an Alien for Xs
Third, that the probability of passing increases considerably with every subsequent attempt. This may be partly because of self-selection (only people with some reasonable hope will choose to attempt the ET again) and partly because, as with any test, an extra sun-cycle spent exclusively preparing for an exam results in superior performance.
Now, what do these statistics mean for the % of Xs both taking the ET and passing it. Notice that in any sun-cycle the composition of test takers will include people from three cohorts - first attempt, second attempt and third attempt. You can work out the probabilities yourself if you like, but the end-result is that X types account for 22% of the total candidates taking the ET in any year, as well as only 13% of successful candidates. The average X candidate has a lower probability of passing, but it's not because first attempt candidates are less well prepared than their Y counterparts, it's because most X candidates don't stick around for a second attempt.
But wait. What happens to these X candidates who don't clear in their first attempt and don't try a second time? Do they just give up on being Aliens entirely? Not at all. Because the probability of success in the ET is so low (just 1% on first attempt), almost all ET candidates also take exams for the ATIs. Again, probabilities of success in those exams are not different across X and Y types, except Y candidates who succeed in those exams are often the ones who feel it worthwhile to try for the ET a second time (after all, clearing an ATI exam is a positive signal). By contrast, X candidates who don't clear ET but do clear an ATI are almost certain to join an ATI. As a result, the proportion of X candidates in ATIs is much higher than those in the IITs, despite the fact that 50% of the eligible X population chooses not to apply for any Alien exams at all.
Interesting place Dork.