Recently there has been a lot of talk and a few lawsuits over the admissions practices of Harvard and the Ivy League schools in general[1]. Broadly speaking, some claim that the admissions process is less of a meritocracy than affirmative action based on race. I’m not going to go into the racism (or possible lack thereof) in the admission process of private instructions, but rather the meritocracy aspect. Harvard and other ‘top-tier’ schools make a concerted effort to establish that their admissions process is fair, balanced, and multi-faceted. However, it has long been suspected that though the admissions process may be these things, it also has an element of randomness to it. Perhaps it was the lack of sugar in the admin person’s coffee that morning, perhaps it was a very poor commute into work that soured someone in the office that day, perhaps there really is a random element to the admissions and they do actually pluck names out of a hat.

I want to focus on that random element and play a bit with some probabilities. Assume that you are actually applying to Harvard and they actually confess to using randomness in their admissions policies. Say that they still require a fee to submit your application. Today that fee is sitting at $75.00. Say that the ‘prestige’ of Harvard remains the same after this hypothetical confession to the use of randomness and that today’s median salary for a Harvard grad stays at ~$85,000.00/year. Got it, okay?

The questions then is: Should you apply to Harvard and hope to be a random applicant that gets in at the $75.00 price? Essentially, I am asking that if Harvard becomes a plain-old Lotto ticket, should you place a bet?

Spoiler: Yes, absolutely.

Let’s go through the math a bit then, shall we? First we need to know the odds of getting into someplace like Harvard. To me, that means an Ivy League school. Some may complain about Stanford, Oxford, UCLA, UT-Austin, and the like. But to narrow down the problem a bit, I’m just going to focus on the Ivies. All the Ivies publish their application numbers, the number of acceptance letters they sent out, and the number of students that actually came to school. Graduation rates are a bit harder to come by and I didn’t include them.

One confound with schools like these, is the ‘legacy’ admission. These are the progeny of graduates, and though all the schools deny that special considerations are given, the school newspapers howl about it constantly. Seeing as a ‘legacy’ admit is not part of the population that are ‘lotto’ players, I have removed them to the best of my ability.

School	Total	Percent	Total Lotto Chance
Harvard			0.030
Applicants	39,506	100.00%
Admitted	2,037	5.16%
Matriculates	1,687	4.27%
Legacy	506	1.28%
UPenn			0.051
Total Applicants	40,413	100.00%
Total Admitted	3,757	9.30%
Total Enrolled	2,457	6.08%
Legacy	393	0.97%
Princeton			0.036
Total Applicants	31,056	100.00%
Total Admits	1,990	6.41%
Total Enrolled	1,314	4.23%
Legacy	202	0.65%
Columbia			0.039
Total Applicants	37,389	100.00%
Total Admits	2,183	5.84%
Total Enrolled	1,456	3.89%
Legacy	0	0.00%
Cornell			0.061
Total Applicants	47,039	100.00%
Total Admits	5,962	12.67%
Total Enrolled	3,375	7.17%
Legacy	506	1.08%
Dartmouth			0.051
Total Applicants	22,033	100.00%
Total Admits	1,925	8.74%
Total Enrolled	1,217	5.52%
Legacy	103	0.47%
Yale			0.034
Total Applicants	31,455	100.00%
Total Admits	1,972	6.27%
Total Enrolled	1,360	4.32%
Legacy	306	0.97%
Brown			0.036
Total Applicants	35,438	100.00%
Total Admits	2,566	7.24%
Total Enrolled	1,829	5.16%
Legacy	549	1.55%
		Average	0.042
		StDev	0.010

Nice! We see that the average chance at getting into an Ivy is about 4.2% (+/- 1%). This is not the worst, it’s between 1 in 20 and 1 in 30, or thereabouts. As far as lotto games are concerned, this is right up there with the odds for a Wednesday night bowling alley lotto.

The next question is then what are the expected returns on a Harvard education. Fortunately, the Washington Post has a great article on the salaries for Ivies[2]. With these numbers, I will be taking the median salary as the expected salary for a random Lotto applicant for the rest of their life. Of course, however, Brown is not cheap to attend at about $71,000.00/year. The cost of 4 years of tuition should be factored in as well.

Note that I am only going off of the expected salary that a graduate can expect to earn from each intuition. This will not include earnings from the stock market with will likely make the expected lifetime gross wealth increase by a substantial amount. Also note that I am using a working lifetime of 38.5 years, the current expected average for babies born in the US in 2001.

School	Median Salary/yr	Yearly Tuition	Total Tuition	Total Lifetime Salary	Net Salary Min. Tuition
Harvard	$85,000	$66,950	$267,800	$3,272,500	$3,004,700
UPenn	$79,000	$73,714	$294,856	$3,041,500	$2,746,644
Princeton	$75,000	$70,010	$280,040	$2,887,500	$2,607,460
Columbia	$72,000	$74,173	$296,692	$2,772,000	$2,475,308
Cornell	$70,000	$76,997	$307,988	$2,695,000	$2,387,012
Dartmouth	$65,000	$73,836	$295,344	$2,502,500	$2,207,156
Yale	$65,000	$69,430	$277,720	$2,502,500	$2,224,780
Brown	$60,000	$71,050	$284,200	$2,310,000	$2,025,800
Avg	$71,375	$72,020	$288,080	$2,747,938	$2,459,858
StDev	$7,664	$3,020	$12,080	$295,058	$298,738

To then determine the price of the ‘lotto’ ticket, you just simply multiply the chance to get into a school by the expected lifetime return:

School	Net Min. Tuition	Chance to get in	Price to Pay to Enter Lotto
Harvard	$3,004,700	0.030	$89,823
Upenn	$2,746,644	0.051	$140,278
Princeton	$2,607,460	0.036	$93,363
Columbia	$2,475,308	0.039	$96,393
Cornell	$2,387,012	0.061	$145,588
Dartmouth	$2,207,156	0.051	$111,595
Yale	$2,224,780	0.034	$74,548
Brown	$2,025,800	0.036	$73,171
Avg	$2,459,858	0.042	$103,095
StDev	$298,738	0.010	$25,707

Wow! College really is worth applying to! Look at the average there! That means that if the price of the ticket for an average Ivy is less than ~$100,000, you should apply. Especially look at Cornell: if the price of admission to Cornell is less than nearly $150,000, you should apply. Remember though, this is based on the assumption that the admissions process is totally random and that they don’t talk to each other.

One question I did get asked is what is the probability of getting into ANY Ivy if you applied to all of them. If the Ivy admissions process is indeed independent from campus to campus, it comes out to just be the sum of all the probabilities: a 33.7% chance. That’s better than 1 in 3! Getting into any HYP Ivy is 9.9%, about 1 in 10.

One thing to look at as well, is the graph of the price of the lotto ticket against the chance of getting into the school:

Look at that! As the chance to get in RISES, the expected maximum price of the Lotto ticket also RISES. The R² is also good at 0.81; I’d say there is a real relationship there! This means that easier to get-into schools are more lucrative for you after graduation. Why this is may be a very hard question to answer, however: perhaps it’s the ‘networking’ effect, perhaps the children of more wealthy parents tend to choose easier to-get-into schools, who knows?

Another thing to take away is that UPenn is the most ‘worth it’ school, relative to all the other Ivies. It has the highest price for the Lotto ticket and the easiest chance to get into, relative to the line of best fit. Essentially, you want to look for the point highest above the line. Conversely, Brown is the least ‘worth it’ school, with the lowest price and a harder chance to get into.

Nonetheless, a $75.00 admissions fee is a pittance to even the price of $73,000 for Brown. That’s about a 1000 to 1 deal on your money!

I think this is why places like Harvard and the other Ivies are SO invested in making certain that their process is NOT random. At the VERY least, they must keep a ‘Wizard of Oz’s Curtain’ of objectivity for their admissions process. If not, then they could be expected to be flooded by 1000 times more applicants for each seat, making the process of admissions a much larger part of the school’s mission. Things would quickly devolve for their ‘brands’ and their graduates would likely have less cache in the world at large. Essentially, all the assumptions I made about salary over a lifetime would evaporate in about a year or two.

In Summary: If you have the extra cash for application fees, apply to every school you possibly can; it’s a great deal as far as betting and ROI are concerned. If you think that your odds of getting into Harvard are better than 1 in 1,197, then mow some lawns and take that shot; it’s worth the cash. Probabilistically speaking, college is worth every single last penny and/or Benjamin.

[1] https://www.cnn.com/2018/06/15/politics/harvard-admissions-asian-american/

[2] https://www.washingtonpost.com/news/wonk/wp/2015/09/14/this-chart-shows-why-parents-push-their-kids-so-hard-to-get-into-ivy-league-schools/?utm_term=.8d16e732aa0d

Update 20180710: I got asked a question on how the other fine institutions of this country stack up against the Ivies as well. So, I decided to take a peek at it. The best data set that I could reasonably find on this was US News and World Report’s eponymous list of colleges and universities. I took the top 25 Universities and the top 11 Colleges (minus a few that didn’t report some stats) and then graphed them all together. Note that some of the earlier analysis that I did was not really applicable to the other schools, so I just took the raw data for granted. Also note that the California schools have substantial discounts for in state and out of state students, however that discount did not count for all that much in the end.

School	Tuition	Room	Salary	Lifetime Income	Accept-Rate	Lotto Price
Princeton	47,140	15,610	66,700	$2,316,950.0	0.07	$162,186.50
Harvard	48,949	16,660	63,100	$2,166,914.0	0.05	$108,345.70
U Chicago	54,825	15,726	54,400	$1,812,196.0	0.08	$144,975.68
Yale	51,400	15,500	60,200	$2,050,100.0	0.06	$123,006.00
Columbia	57,208	13,618	62,200	$2,111,396.0	0.06	$126,683.76
MIT	49,892	14,720	76,900	$2,702,202.0	0.08	$216,176.16
Stanford	49,617	15,112	70,300	$2,447,634.0	0.05	$122,381.70
Upenn	53,534	15,066	65,000	$2,228,100.0	0.09	$200,529.00
Duke	53,744	15,500	62,700	$2,136,974.0	0.11	$235,067.14
CalTech	49,908	14,796	78,400	$2,759,584.0	0.08	$220,766.72
Dartmouth	52,950	15,159	61,700	$2,103,014.0	0.11	$231,331.54
John Hopkins	52,170	15,410	61,600	$2,101,280.0	0.12	$252,153.60
Northwestern	52,678	16,047	56,400	$1,896,500.0	0.11	$208,615.00
Brown	52,678	14,020	57,500	$1,946,958.0	0.09	$175,226.22
Cornell	52,853	14,380	63,100	$2,160,418.0	0.14	$302,458.52
Rice	45,608	13,850	63,300	$2,199,218.0	0.15	$329,882.70
Vanderbilt	47,664	15,584	58,000	$1,980,008.0	0.11	$217,800.88
Nortre Dame	51,505	14,890	60,400	$2,059,820.0	0.19	$391,365.80
Wash U St. Louis	51,533	16,006	57,300	$1,935,894.0	0.17	$329,101.98
Georgetown	52,300	16,036	57,800	$1,951,956.0	0.17	$331,832.52
Emory	49,392	13,894	54,600	$1,848,956.0	0.25	$462,239.00
Cal	42,112	17,274	61,800	$2,141,756.0	0.16	$342,680.96
UCLA	41,270	15,441	55,000	$1,890,656.0	0.18	$340,318.08
USC	54,259	14,885	56,700	$1,906,374.0	0.17	$324,083.58
Carnegie Mellon	53,910	13,784	69,700	$2,412,674.0	0.22	$530,788.28
Williams	53,550	14,150	55,500	$1,865,950.0	0.18	$335,871.00
Bowdoin	51,848	14,132	51,700	$1,726,530.0	0.15	$258,979.50
Wellsley	51,148	15,836	48,400	$1,595,464.0	0.15	$231,342.28
Middelbury	52,496	14,968	51,600	$1,716,744.0	0.16	$274,679.04
Pomona	51,075	16,150	52,100	$1,736,950.0	0.09	$156,325.50
Carleton	52,782	13,632	49,100	$1,624,694.0	0.23	$373,679.62
Claremont McKenna	52,825	16,220	59,600	$2,018,420.0	0.09	$181,657.80
Davidson	49,949	13,954	49,500	$1,650,138.0	0.20	$330,027.60
Washington and Lee	50,170	11,730	57,300	$1,958,450.0	0.24	$470,028.00
Colby	53,120	13,660	50,300	$1,669,430.0	0.19	$317,191.70
Colgate	53,980	13,520	56,700	$1,912,950.0	0.29	$554,755.50
Average	51,167.83	14,914.44	59,350.00	2,020,645.89	0.14	$275,403.74
STD	3,264.00	1,108.99	7,091.38	274,462.62	0.06	$113,906.42

 

This data is not all that informative in the end, but the graph of it should be a bit easier to digest.

Well, that’s a bit of a mess with the labels, huh? I tried to get them to look better, but it ends up being even worse. Try messing with the data yourself a bit if you are interested as well. The take away is that, again, you should pay more for the lotto ticket to get into easier to get into schools. By far, the best school to do that is Colgate at nearly $555,000.00! At a $60.00 application fee, it is discounted at 1 to 9,245.93, an incredible steal. However, the schools that are best ‘worth it’ relative to the other elite schools are Carleton and Davidson, as they are the furthest under the line of best fit. A surprise is Carnegie Mellon, which is far and away above the line of best fit. This means that the school’s chance to get in is relatively low for the lotto ticket price, relative to the other elite schools. As an aside, if you were to apply for all these schools, you’d expect to get into very close to Five of them.

Again, go to college kids. The ROI is insane.