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<p>And I will reply that examples with dice (which in this case looked remarkably like coins 
) are wholly inadequate to model college admission decisions. </p>
<p>Are you checking any of your posts with a statistics teacher at your school? That could be the basis for a very interesting conversation in real life.</p>
<p>tokenâs right. I just aced a probability unit test in AP stats this week, so that gives me credibility right? lol</p>
<p>Readers of this thread are welcome to check their conclusions with knowledgeable members of college admission committeesâalas, all probably very busy at the momentâor professors of statistics as they wish. There are also independent scholars of the college admission process, often professors of economics or education, located at many universities. </p>
<p>1) The OPâs statistical model of college admission, which is posted from time to time here on College Confidential, is not a valid âproofâ in statistics and is not a correct model of the college admission process. The error in the model consists mostly of assuming that an individual studentâs admission results from each college will be âindependentâ in the statistical sense, which is a prerequisite for applying the multiplicative rule to the problem. An additional problem with the model is the assumption that group base rates of admission can be attributed as âprobabilitiesâ to individual applicants. Both errors have been mentioned and discussed in several thoughtful replies above. </p>
<p>2) A student with </p>
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<p>could quite possibly be rejected by ALL of {Harvard, Yale, Princeton, MIT}, and it would not surprise me at all to hear of such a case. (I think we do hear of such cases each year here on CC.) Issues other than the test scores and grades specified matter in the admission process. We can all think of extreme cases (the convicted felon, the person with grotesque body odor during interviews, the person caught plagiarizing AP articles in the school newspaper) who would be rejected, whatever the numerical stats. And most people reading this thread can readily agree with the examples given of admission factors that matter enough to the HYPM admission committees that they could fill up a class with students such that there is simply no room to admit every student with the specified numerical stats. </p>
<p>3) And yet if we look beyond the HYPM mentioned in the thread title, and consider all eight colleges in the Ivy League, the freshman class sizes look like this, from smallest to largest (size of entering freshman class in fall 2006):</p>
<p>Dartmouth College 1081
Princeton University 1228
Yale University 1315
Columbia University 1337
Brown University 1469
Harvard University 1684
University of Pennsylvania 2385
Cornell University 3238 </p>
<p>Thatâs more students than reach the SAT scores specified by the OP (leaving aside entirely the issue of high school grades), so it is CERTAIN, by the pigeonhole principle, that at least one Ivy League college will admit at least one student with lower stats than those specified by the OP. Then the interesting question becomes, what did that student do to look good to an Ivy League college? How can someone else do that? </p>
<p>4) There surely is some advantage in applying to more colleges rather than fewer, as long as you are applying to colleges you would really like to attend. For sure you canât be admitted by a college you never apply to at all. </p>
<p>5) Colleges within the Ivy League, and their several âpeerâ colleges, are all very selective compared to most colleges nationwide, but they vary considerably compared to one another in selectivity. </p>
<p>What else can we agree on here?</p>
<p>It would be nice if you would stop dodging the substance of my arguments.</p>
<p>I am not using dice or coins to model the college admissions process, and whether or not they can accurately model the college admissions process is irrelevant. I have not argued that they are good for modeling the college admissions process.</p>
<p>What I am arguing is that your understanding of independence is incorrect. You implied that if the outcomes of past events can be used to update our opinions on the probabilities of future outcomes, then the events are not independent. I am only using the dice example to show why this reasoning is flawed.</p>
<p>I have doubts about that the expertise of that statistics teacher you quoted. Try to find in a print source or on the internet where it says that if the outcomes of past events can be used to update our opinions on the probabilities of future outcomes, then the events are not independent. You are not going to find it because its not true.</p>
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<p>Because two threads have been going on about this issue, Iâm not sure which thread has included a prominent posting of this reply I received about the issue when I took the issue (with the OPâs permission) over to the AP statistics teacher email list. </p>
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<p>Now this is a free country, and anyone is free to disagree with any teacher he or she pleases. But I will argue strenuously, and I hope politely, that if your observation of previous events leads you to âpredictâ the outcome of the next event, then the events are not independent. If there is correlation from one event to the next, there is not independence. </p>
<p>I ask again, are you checking your posts here with any statistics teachers?</p>
<p>I donât need to check my posts with a statistics teacher.
You have been through all the Statistics classes, as have many other people on this board.
If I am indeed incorrect, there should be more than enough people knowledgeable enough in Statistics to not only point out that I am wrong, but also demonstrate using logic and arguments why I am wrong.</p>
<p>âBut I will argue strenuously, and I hope politely, that if your observation of previous events leads you to âpredictâ the outcome of the next event, then the events are not independent. If there is correlation from one event to the next, there is not independence.â
Nowhere on the internet have I found a website stating that this is true. And its kind of just taking your word for it, since you really havenât shown has this results from the definition of independence.</p>
<p>There is a difference between being able to âpredictâ or guess as to the probabilities of certain outcomes</p>
<p>and</p>
<p>definitively knowing that the probabilities of certain outcomes have changed.</p>
<p>Once again, knowing for sure the probabilities of certain events is different from gaining some new information as to what the probabilities might be.
Knowing that Cornell has rejected Bob does not change the odds that Harvard will admit Bob, although it may make you think that Harvard will be less likely to admit Bob.</p>
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<p>Knowing that Cornell has rejected Bob makes us enter a different base rate of acceptance into our calculation about Bobâs chances at Harvard, which amounts to the same thing. But what exactly are âoddsâ of being admitted to college?</p>
<p>One disadvantage of carrying on a discussion in two threads is that active participants have to post to both threads to make sure everyone is well informed. A very important article by a professor of statistics </p>
<p>[Advice</a> to Mathematics Teachers on Evaluating Introductory Statistics Textbooks](<a href=âhttp://statland.org/MAAFIXED.PDF]Adviceâ>http://statland.org/MAAFIXED.PDF) </p>
<p>illustrates the problems many college-educated people have in understanding statistics because they have been taught statistics by nonstatisticians. I enjoy reading this articleâthe author has a good sense of humorâand I enjoy thinking about the issues in this article, because I have discovered, to my dismay, that many public library systems have NO good books about statistics and what statistics is, and many college students who have taken statistics courses have missed out on important issues of statistical reasoning. There is a lot of food for thought in the article. The AP statistics syllabus and âreformâ statistics courses in universities are attempts to overcome the problems in statistics education described by the article.</p>