Probability/Chance up if applying more Ivies schools?

<p>You don’t need to know your exact probability to develop an effective application strategy.</p>

<p>The way we worked our strategy for our D was relatively simple:
-Any school for which she was in 25-75th SAT percentile was considered a reach.
-Any school for which she was in the 75th-90th percentile was considered a match.
-Any school for which she was above the 90th percentile was considered a safety. </p>

<p>At no school was she below the 25th percentile. </p>

<p>We roughly estimated:
In the first group she may stand a 10% chance.
In the second group around a 50% chance.
In the third group around a 99% chance. </p>

<p>We then allocated the applications accordingly:
10 in the first group
5 in the second group
2 in the third group. </p>

<p>Results came in as follows:
-3 admissions in the first group (30%)
-3 admissions in the second group (60%)
-2 admissions in the last group (100%)</p>

<p>

</p>

<p>That sounds rather tongue in cheek. I don’t look at the chances threads on CC, in whatever forum they appear, much at all, but my general impression is that they tend to provide lowball estimates of chances and scare applicants unduly. Everyone should be scared enough to line up a safety, first of all, but no one should be dissuaded from applying to a highly desired college at which the applicant has some reasonable chance because of some applicant characteristic.</p>

<p>I was kidding. The chances threads include both overly optimistic and overly pessimistic evaluations of other people, often provided by people with no better sources of info than the people asking for the chances. I think the approach cellardwellar outlines is pretty good. 17 schools is a lot, and it may be too many apps for some kids to manage effectively. But I agree with the upside down pyramid.</p>

<p>Okay, I know this is not scientific, but here it goes: I read most of this thread yesterday and then went up to my DD’s high school for a previously scheduled parent-teacher conference with the college counselor. (My DD is an excellent student who tests very well). I asked, “So, to how many schools do you recommend my DD apply?” and the Counselor responded, “We recommend two in-state schools plus eight out-of-state schools, but since your DD is ‘Ivy competetive’ we recommend that she apply to more.” Now, I didn’t laugh out loud, but I was thinking of all you guys!</p>

<p>QuantMech: I understand what you are trying to say, that the process is not random as everyone in the pool does not have the same chance of acceptance, some who in reality have no chance. Within the pool of applicants, there is smaller pool of strong candidates, among whom there is more randomness i.e. some candidates in this pool may be rejected while some people with 'slightly lesser track records" may get accepted. In the weaker pools, everyone gets rejected. Again it is not completely random in the stronger pools, but there is a larger element of randomness.</p>

<p>In other words there is a stratification of the pool, and the degree of randomness depends upon the particular strata in which the candidate fits. As all the ivys are looking for similar but not exactly the same things (good grades, high SAT scores, strong EC’s great recommendations), the strong strata would be fairly similar between one college and another. There are going to be some variations within the ivy’s (there are the lower and higher ivy’s, so a strong candidate at Brown may not be strong at Harvard). There is also a great degree of subjectivity i.e. introduction of errors due to bias.</p>

<p>So your contention that a mere multiplication of probabilities may lead an overstatement of actual probability as many of the noise factors have not been removed. I agree with that. However going back to OP’s question, I stand by these statements</p>

<p>1) Applying to more colleges always increases your odds, but in most cases the increase is not significant. Increasing your odds from 1% to 1.2% will not get you admission to a top institution.</p>

<p>2) If you are in the strong zone (however you describe it), it always is to the advantage of the student to apply to more than one ivy. It increases their odds as schools are looking for approximately the same things and if one is rejected due to randomness at one school, the chances are that the randomness will not work against that the student in another school. I believe this what you have been saying also.</p>

<p>I agree with you, mazewanderer. </p>

<p>I think that when an applicant can reasonably guess that his/her personal odds are about double the raw odds, then it starts to be really worthwhile to apply to multiple top schools (the ones that make sense for that applicant). This, however, is not mathematical; it’s just based my own risk/benefit analysis. If an applicant with true odds of about 5% at an individual top school has the time and money for a large number of applications, and wants to apply to a large number of schools to gain a relatively small improvement in the odds, why not?</p>

<p>Quantmech.
I agree that your suggestion is a good heuristic or thumb rule, i.e. if an applicant’s odds are better than average applicants odds (twice as good may be a good starting point), then it makes more sense to apply to more schools if they have the interest time and money.</p>

<p>Also, why limit to 5%, even a person with 1% odds can apply to as many schools as time and money permit. It does not affect the strong applicants chances very much. And that is what schools are trying to do, increase the applicant pool though direct and subtle marketing. A lot of this is driven by rankings (USNWR being the prime culprit). If there was less wild hope in the applicant pool, the pool would be stronger and selection decisions more random.</p>

<p>Just remember that even though a blind pig finds an acorn every once in a while, it only happens if he’s looking in a part of the forest with acorns.</p>

<p>Perhaps another way to think about the OP’s original question is: under what scenarios is there a considerable downside to also applying to “extra” highly selective colleges? Say you’ve targeted a group of colleges that you think are a good fit for you (with a few ivies, a few other highly selectives, some safeties, etc.) why not also apply to the rest of the ivies anyway? </p>

<p>1.Many posters have already mentioned that writing additional apps and the time/effort involved will leave less time to focus on the apps that might matter most to you, and dilute the quality of those apps. In the previous discussion many were willing to ignore this assuming an applicant has unlimited time (and $ for app fees), so I’ll move on to some other downsides.</p>

<p>2.The number of schools applied to, and the mix of them, will affect the guidance counselor’s letter. This is IMO the big downside. While applying to more schools may be in the individual applicant’s interest, the GC has different goals, and (s)he needs to maximize the admissions success of all students at the school, both this year and in the future. If 5 kids who don’t care much about Dartmouth apply to all ivies, that mucks up the chances of that other kid from the HS who really wants Dartmouth standing out. I think of the analogy to prescribing antibiotics - the patient always wants to get it early, but docs have a public safety benefit to consider that if antibiotics are over-prescribed then drug resistant strains of the disease develop more quickly. If the GC strongly suggests applying to 8 schools and you insist on 15 against their advice, don’t you think it might affect the letter? </p>

<p>Even if you are not directly contradicting their advice, apply to “extra” schools can impact the letter. While you can dial up or down the emphasis on your various strengths for different schools, the GC is likely to write basically 1 version for all the schools, and I believe it is disquieting for adcoms if what you write and what the GC writes don’t resonate completely with each other. GCs may also have a harder time conveying you as someone who knows your strengths and where you are going if they feel you are taking a more scattershot approach to applications. And perhaps most importantly, GCs can’t signal you as “likely to enroll if accepted” if you apply to lots of schools.</p>

<ol>
<li> While I agree the admissions decisions are independent (e.g., Yale does not change their decision based on the decisions of other schools to admit you or not), I can easily see how knowing where else you have applied might affect their decision on whether or not to admit you. Mostly, I think this is a question of authenticity - does the range of places you apply to make sense given the person you claim to be on your application? If your essay to Yale talks about your commitment to community service with teaching math to inner city kids that you want to continue in college, it makes sense for you to apply to Columbia or Penn, but if you apply to Dartmouth is that still believable? If you say you are attracted to Princeton because of its size, focus on undergrad education and suburban location, is that believable if you’ve applied to every ivy plus U Chicago, Michigan, etc.?</li>
</ol>

<p>Also, I do think yield matters to them to some extent, and while they expect kids to apply to a reasonable range and number of colleges they might be less inclined to admit someone they thought was simply “fishing” by applying to all the ivies.</p>

<p>How do they know where else you have applied? I think they have some knowledge in some cases, though clearly it is imperfect and not universal. Some kids put this on facebook, some blab this to interviewers (alumni or school), some GCs might intimate this in letters. I also heard a financial aid expert mention that information about where else you are applying is conveyed with financial aid forms within the ivies, but I don’t know how true or widespread that is - maybe someone who knows better can chip in.</p>

<p>chandelle: You have made some good points on why applying to many colleges may not be effective. </p>

<p>I think the issue here is the law of diminishing returns. As you apply to more colleges, there is a point where the effort of more applications is not equal to increase in chances of admission. This point of inflection is student specific: Students own records, ability to afford the cost and time, the impact on recommendations etc.</p>

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<p>Your reply is ambivalent. Let’s get to the mathematical heart of the question:</p>

<p>Do you agree or disagree that any probability model of (one individual applicant’s) admission that is NOT multiplicative is invalid? Here “multiplicative” means that probability of any combined outcome such as Prob[Joe is admitted to Stanford, rejected by MIT, waitlisted by Harvard] is equal to a product of the per-school probabilities, in this case Prob[Joe admitted to S] x Prob[Joe rejected by M] x Prob[Joe waitlisted by H].</p>

<p>Several notes:</p>

<ol>
<li><p>The word “invalid” above is a euphemism. More honestly one might call the non-multiplicative models “embarrassingly wrong”, “bogus”, or “professionally disqualifying if advocated by a statistics teacher” (as in tokenadult’s FAQ).</p></li>
<li><p>Assume for purposes of this question that all applications are ordinary, regular-round (deadline around January 1, April 1 notification), and there is no athletic recruitment or “special handling” category applicable, that might trigger exchanges or monitoring of information about Joe among the universities. Joe is not the POTUS’ nephew, and he has not been on any school’s radar prior to applying. He is a generic applicant who, from the admissions point of view, is fungible with many others as far as his qualifications and each university’s level of interest in him are concerned.</p></li>
<li><p>I am not assuming that probabilities are necessary represented by a single number. One might quantify Joe’s odds as a range of probabilities (“between 3 and 99 percent chance of admission to Harvard”), or some Bayesian odds-of-odds measure (“equally likely to be any probability between 3 and 99 percent”), or probabilities that vary over time and as a function of the stock market and the weather in Boston. One can still multiply these fancier sorts of probability data. Any such model is fine, but the issue is whether it must satisfy multiplicativity in order to pass the laugh test.</p></li>
</ol>

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</p>

<p>If your odds (of at least one acceptance) are 1 percent at a one institution, they are essentially quadrupled by applying to four similar institutions. Probabilities are close to being additive in the low range. For applicants who calculate rationally that one longshot is worth it (they are not just sending a lottery ticket to Harvard because having that longshot is a goal in itself), additional longshots are worth it for the same reason.</p>

<p>It is only at the very high probabilities that the odds improvement is marginal, and in those cases, it isn’t the odds that matter; one wants to have multiple offers, not a single acceptance. Conclusion: send more applications.</p>

<p>In the moderate probability range, sending more applications has the greatest chance of making a difference between having some acceptances and having none. Conclusion: send more applications.</p>

<p>This uniformity of strategic implications reflects the fact that expected number of acceptances is a better measure of what applicants are aiming to improve. In all probability ranges, the expected number of acceptances is (crudely) proportional to the number of applications. Double the number of schools, double the expected number of acceptances. </p>

<p>Given this, and the extent to which one might over- or under-estimate the probabilities, for purposes of this question it almost makes no difference to strategy whether one can guess the probabilities correctly. It is hard to think of a scenario where (subject to the obvious non-mathematical externalities) it makes sense to send one application and not a much higher number.</p>

<p>siserune, yes I agree that the multiplicative approach works for an individual applicant, with the applicant’s personal probabilities pH, pY, and pP. In the context of this thread, I think I am the one who suggested it.</p>

<p>When I said that the odds would not go up for most applicants as much as one would predict if the admissions processes were modeled as independent, random selections, I meant: if applicant X takes the raw odds, rH, rY, and rP, computed by dividing the number of acceptances by the number of applications, and then computes (1 - rH) (1 - rY) (1 - rP) as the odds of getting into none of H, Y, and P, X will be overestimating X’s own chances, in most cases. This is due partly to the presence in the pool of the special-category applicants you mentioned in #171, note 2. This alone makes the raw odds higher than the actual odds for a generic applicant. But additionally, in saying “most,” I am estimating the number of “fungible” applicants (your term in #171) as double to triple the number in the admitted class, but not much higher, based on admissions-office rhetoric of the type, “We could easily have admitted a second class that was essentially indistinguishable from the class we selected.”</p>

<p>With regard to #172, you make a good point that applicant X does not need to know X’s true values of pH, pY, and pP to know that increasing the number of applications will always give better odds of an acceptance–unless the added app has zero probability of success or an app already submitted has p = 1–and in the latter case, by submitting additional applications, X is still likely to increase the number of acceptances, as you pointed out.</p>

<p>I think chandelle has made a good point in #169, though, with regard to the OP’s question–if the GC is advising the student not to apply to more colleges, and if the GC is writing the recommendations, and and if the GC will be irritated at having to fill out more forms, where is the break-even point of increased odds with multiple apps vs. decreased odds with an annoyed GC?</p>

<p>The GC point might be a good one at some schools. It probably would not be a big factor at my kids’ high school, where I’m pretty sure the GCs are too busy to engage in that level of analysis. They write the rec (based on the brag sheet), and photocopy it the requisite number of times.</p>

<p>Besides looking at each college’s SAT (or ACT) interquartile range, e.g., </p>

<p>[College</a> Search - California Institute of Technology - CALTECH - SAT®, AP®, CLEP®](<a href=“College Search - BigFuture | College Board”>College Search - BigFuture | College Board) </p>

<p>[College</a> Search - Massachusetts Institute of Technology - MIT - SAT®, AP®, CLEP®](<a href=“College Search - BigFuture | College Board”>College Search - BigFuture | College Board) </p>

<p>what else would help an applicant gauge chances for admission at a particular college?</p>

<p>^^^
I like Caltech’s mid 50% Math SAT range.
In a couple years the range is going to be from 800 to 800.</p>

<p>:)</p>

<p>

The Naviance scattergram for the applicant’s high school, if it’s available. That’s one way of getting a little insight into how the college views the GPAs from that particular high school.</p>

<p><a href=“QuantMech:”>quote</a> </p>

<p>siserune, yes I agree that the multiplicative approach works for an individual applicant, with the applicant’s personal probabilities pH, pY, and pP.

[/quote]
</p>

<p>Great, but the posted question was much more specific: is that approach the only viable one — does it completely exhaust the universe of probability models that could possibly apply here, with the non-multiplicative models being nonsense? </p>

<p>Given your username I’m sure you understand exactly what the issue here is. Any model that allows correlations (i.e., non-independence of the different schools’ decisions on a single candidate) requires “spooky actions at a distance”, such as Princeton telepathically determining the admission status of Harvard candidates.</p>

<p>That actually happened a few years ago, by means that are less magical though just as entertaining. Similarly, the financial aid decisions used to be explicitly correlated until the US Dept of Justice stopped it on anti-trust grounds. Telephones and e-mail really are a form of action at a distance. But it is not likely that any of these processes are utilized for any typical applicant this decade.</p>

<p>

</p>

<p>Indeed, there isn’t any other option available to be suggested. That’s the point. I further claim that this lack of options should be totally clear to anyone with minimal technical training in probability and statistics, and it is amazing that an AP Stats teacher would argue against independence (in what was clearly a one-applicant context) on a stats discussion list without being contradicted. </p>

<p>I’ll address your other comments about independence separately, but given that it’s been several years and dozens of repostings of tokenadult FAQ claims that the multiplicative probability calculations are “wrong”, we should make clear what is and is not the correct mathematics here. I don’t dispute that there are other sets of random variables floating around the discussion that are not independent, but it’s a separate question from the can-you-multiply matter.</p>

<p>siserune, in response to your question whether the multiplicative approach is the only viable one for an individual applicant who is working with pH, pY and pP (as opposed to rH, rY, and rP): yes. I think so. </p>

<p>Still, there are assumptions that go into this. I am assuming that there is a probabilistic aspect to the decision-making, even after all of the differences in the strength of the applications have been taken into account. I base this assumption on my experience on selection committees, though not in an Ivy-admissions context. But it’s possible that the whole process is entirely deterministic. I am also assuming that each college makes its decisions without knowledge of any other college’s decisions. I’m quite a bit more confident of this. There may be other assumptions that I haven’t noticed yet.</p>

<p>I’m usually comfortable calling a string of equations “nonsense,” if they are. I’m a lot less comfortable calling an English-language discussion of a mathematical idea “nonsense,” because issues can arise in the math-to-English conversion, and there might be an interesting idea behind the words.</p>

<p>I am going to make some statements here:</p>

<p>1) The probabilities of acceptance (or rejection) to a top school are not totally independent, there is some degree of correlation between the probabilities. Also, even though the admissions committees may not talk to each other, in some cases they would be able to gauge the students attractiveness to other colleges. Washington University at St. Louis and some other schools (The Tufts syndrome) are reputed to reject candidates who they think will go a higher rated institution. Even though this causes errors (rejecting a good candidate), this could happen in any institution in individual cases.</p>

<p>2) Even with this the multiplicative approach is the correct statistical approach. Statistically the confidence interval (in laypersons terms though not really correct, the margin of error) will be very wide. Generally one would expect that multiplicative approach will lead to over stating of the results, and you will not have confidence in the results due to the introduced bias and the wide margin of error.</p>

<p>3) There is so much of subjectivity and unknowns in the admission process that one would never be able to estimate the probability itself i.e. the probability of admission itself is an estimate, which compounds the error. </p>

<p>4) Hence all you can derive from basic calculations are thumb rules. If you are in High and the medium zones (your probability of acceptance is say higher than the average, twice as high being a cut off point for those who are looking at statistical odds) it makes sense to apply to more institutions. The lower your perceived probabilities, applying to multiple institutions does not provide an noticeable advantage. This is what the OP was looking for.</p>

<p>siserune in post 172 you state that if </p>

<p>

</p>

<p>The improvement may be marginal but it still makes sense. If you have a 90% probability of acceptance in two schools, you have 10% probability of being rejected if you apply to one school. If you apply to a second school, you drop your probability of rejection from both schools drops to 1%. Again, this is assuming a lot of the independence stuff we have talked about, but it still makes sense to apply to say two schools or three schools instead of 5 schools. If you are the 90% level applying to 5 schools as opposed to 4 schools will increase your probability marginally as you have stated. At the 20% probability of acceptance level, 5 or 6 schools may make more sense. </p>

<p>At the low level, probabilities are still multiplicative but appear to be close enough to considered additive. At 1% level applying to 3 schools increases your probability to 2.97% (close enough to 3) but in my opinion not enough to get into an ivy league. This does not mean one should not apply, it just says the odds are against you.</p>