Re: correlated Vs independent. The confusion is in the meaning of the term “independent”. In statistics, “dependent” does not mean that one affects the other. So if there is a third variable which affects the two variables at which we are looking, that makes these two variables dependent, even though they do not directly affect one another. So the price of gas and the height of a tree in your background are not really affecting each other, but they are not statistically independent, since they are both affected by time.
So, since acceptance at any college is affected by stats and EC, that means that probability of acceptance at colleges which have similar requirements are not independent of one another. However, there is no causal relationship between acceptance at one college and acceptance at another college. So the chances of a kid being accepted to Yale are not dependent on whether they were accepted to Harvard, even though kids with high stats and very good ECs have a better chance at being accepted at either.
What we would need to do is to use the joint probabilities of stats and acceptance for each college. Imagine somebody buying multiple tickets for multiple lotteries. For every lottery, the probability of getting a prize increases with the number of tickets bought. If you compare two individuals, and one has bought 10 tickets from each lottery, while the other has bought 100 tickets from each lottery, the person who bought 100 tickets has a larger chance to win a prize from at least one lottery, and has a higher chance at each lottery. Now, the number of tickets bought at each lottery is correlated with the chance of winning a prize at that lottery, so, just like out students, we can say that the chances of the person winning is not independent of the number of tickets they bought.
But one can calculate the total chance of getting a prize by multiplying the probabilities of receiving a prize from each lottery, given the number of tickets that a person has bought for that particular lottery.
You can do the same for acceptances to colleges - you multiple the chances that an applicant has to be accepted at each college, given their stats and profile. That will provide the probability that they will be accepted to at least one college.
Unfortunately, there is no real way to calculate these chances ahead of the admissions season. First, they are dependent of the applicant pool, which differs year to year. Second, they depend on the AOs. Third, they depend on the order which applications are reviewed - the chances of an individual may change depending on whether they are #35 versus #1,394. And so forth, and so on.
Then there is the fact that there is no linear relationship between stats and acceptance. The probability of acceptance does not increase linearly with GPA, SAT, number of ECs, engagement in ECs, number of leadership positions in ECs, number of awards, the level of awards, etc. Add to that the fact that the distribution of GPAs, SAT/ACT scores, and EC activities are also not linear.
What this means is that there are a bunch of applicants whose probability of acceptance to, say, HYPSM, is way higher than 50%, which is why they get multiple acceptances. However, it must be noted that they do not usually get accepted to ALL of the low acceptance colleges, which is what one would expect from a probabilistic process.
However, since the effects of GPA, SAT/ACT, ECs, and other factors can change from year to year, it is impossible to actually predict anything except very very crudely. And all we can do is provide very rough estimates, like “reach”, “target”, and “safety.”
So, @alejaz, while it is, theoretically, possible to calculate that, if we know what the chances will be for kids with your stats at each of the colleges you will mention, we cannot actually know those probabilities. What we can tell you, based on what we have seen in the past. is that, with those stats, even after applying to 10 colleges, you will almost certainly be rejected by all of them. That does not mean that you should not apply to any, but that you should not put the effort and money it would take for you to submit 10 applications. Your chances of being accepted with an application you churn out in an hour is so low that it isn’t worth the hour you took to write it.