Wow. It’s amazing how so many people on this site are having so much trouble with a relatively simple mathematical concept. I’ll try once more to explain some things and respond to people.
@Lindagaf: First off, what’s been covered in this thread is not about turning the entire process into a bunch of numbers.
As a bit of an aside, because I don’t want to sidetrack things: In fact, almost anything can be modeled mathematically, to a certain degree of accuracy. Further, it’s almost impossible not to think about it mathematically, whether you’re doing it consciously or not. You’re doing it. What’s essential to realize, though, is that even if you argue that modeling it mathematically is not effective or accurate, basic mathematical concepts (like statistical independence) are still valid.
Back to this thread. It’s really been about two things: Does a top student applying to multiple top schools increase their chances of getting into at least one of them, and are admissions events independent.
On that first question, consider this: Does a student applying to two schools have a greater chance of getting into at least one school than if they had only applied to one school? Don’t overcomplicate this, that’s what it really boils down to. And, other than some exceptional circumstances, the answer has to be yes. How can it be otherwise?! It’s got nothing to do with whether you’re modeling the whole process mathematically.
One exceptional circumstance is if the decisions at the two schools are perfectly correlated. (BUT BY THE MATHEMATICAL DEFINITIONS, THAT MEANS THAT THEY AREN’T INDEPENDENT. [ALLCAPS used intentionally.]) This is possible, but highly unlikely. Think of what you’re saying – that the decisions at schools are somewhat random and unpredictable (which, BTW, is thinking about them mathematically). Which means that the decisions at two different schools are likely to be based on different factors. Which should lead you to the conclusion that applying to multiple schools increases the chances of getting into at least one of them. To put things the way you’re phrasing them, you might hit one committee on a bad day and another on on a good day.
On to the second question:
You really think mathisfun and mathgoodies are “qualified” and “precise” (let alone “very, very precise”?! I gave you the one true pure precise mathematical definition of probabilistic independence: P(A.B) = P(A) * P(B). That’s it. Look it up in a probability/statistics textbook, and you’ll see that’s all it is.
So tell me, how do you get from that to anything involving some events occurring before another, or some events affecting another? You can’t. They’re not there. They only come from thinking about it intuitively and imprecisely.
Show me, using this formula, how you can justify believing that someone who gets into a top school is very likely to get into a lesser school, but yet the two events are independent.
I’ll show you the opposite. A corollary of the above mathematical definition is that if A and B are independent, P(A|B) = P(A). That is, the conditional probability of A given B is the same as the unconditional probability of A. Let’s say B is getting into that top school and A is getting into the lesser school. This corollary says that the likelihood of A, getting into the lesser school, is not changed if A, getting into the top school, is true. But that’s inconsistent with believing that someone who gets into a top school is very likely to get into a lesser school. Which means we’ve reached a contradiction, so the two events are not independent.
I’ll reiterate one thing, because it seems to be repeated multiple times in this thread. Many people are saying that the admissions decisions at top schools correlate fairly strongly. This is probably true. BUT THAT MEANS THEY AREN’T (MATHEMATICALLY) INDEPENDENT. It doesn’t matter how the decisions are made, any degree of correlation means not independent. Don’t confuse the everyday meaning of independent with the mathematical definition.
“This is both wrong, and rude. You should apologize to the person you posted this to for this. No one has called you stupid. It’s uncalled for.”