"Congratulations" is the single greatest word to begin an email. For a graduating high school student, it's immensely more exciting. You deserve not only to be accepted into a university, but also to be simply congratulated for your efforts. To relieve some anxiety about your future, I'll offer a different, more optimistic perspective on the chances of seeing that word in an email from the colleges you hope to be admitted to.

Background

I'll let our overlord known as the College Board define the categories in which I will group the types of colleges: the easily-attainable "safeties," the usually-achieveable "matches/targets," and the you'll-never-be-accepted-no-matter-how-hard-you-try "reaches."

I'll define the acceptance rates of each category by these percentages that I just made up, and then to simplify this simple blog post even more, I'll take the average percentage within each range to use in the calculations:

• Safety: 75 - 100% → 87.5%
• Target: 35 - 74.9% → 55%
• Reach: 0 - 34.9% → 17.5%

* These percentages are based on almost nothing and don't take into account any real factors that may affect admissions. They only generalize the overall acceptance rates of colleges.

"And" probability

P() represents the probability of whatever is within the parenthesis. If we want to find the probability of two independent events happening at the same time, we multiply the probabilities.

P(rolling a two on a six-sided die AND rolling a four on another six-sided die)

= (1/6) * (1/6)

= (1/6)^2

= 2.78% chance

Using probability for college acceptances

We can use "and" probability to find the chance of being accepted into multiple colleges, assuming the events are independent.

The acceptance rate of Purdue University is around 53%, New York University is around 12%, and Georgia State University is around 67%.

P(accepted into Purdue University AND accepted into New York University AND accepted into Georgia State University)

= 0.53 * 0.12 * 0.67

= 4.26% chance

Using probability for college rejections

If we want to find the probability of being rejected from all three universities, subtract the acceptance rate from 1 to find the rejection rate, then multiply them together.

P(rejected from Purdue University AND rejected from New York University AND rejected from Georgia State University)

= (1 - 0.53) * (1 - 0.12) * (1 - 0.67)

= 13.6% chance

Using probability for the bare minimum

If we want to find the probability of being accepted into at least one of the three universities, subtract the probability of being rejected from all three from 1.

P(accepted into at least one of the following: Purdue University, New York University, or Georgia State University)

= 1 - P(rejected from Purdue University AND rejected from New York University AND rejected from Georgia State University)

= 1 - ((1 - 0.53) * (1 - 0.12) * (1 - 0.67))

= 86.4% chance

We can't use "or" probability here by simply adding the probabilities because that would give a number greater than 1, which doesn't make sense in this case.

Using the binomial distribution

If we want to find the probability of being accepted into at least two, at least three, or at least any number of colleges, we can use the binomial. Instead of showing the entire calculation, I'll just use the binomcdf() function on my favorite and only graphing calculator, the TI-84 Plus.

A limitation of the binomial is that there must be only one probability. We can use it to find the probability of getting accepted into at least two of five colleges all with the same acceptance rate, for example, but not when the acceptance rates of those colleges differ. If you want a thorough explanation of the binomial, this is a good one.

P(accepted into at least two of five colleges that all have an acceptance rate of around 40%)

= 1 - binomcdf(5, 0.40, 1)

= 66.3% chance

Disclaimer

Although luck is a significant factor, acceptances are not entirely random chance as these calculations may suggest. These calculations only generalize overall acceptance rates of colleges into three basic categories, which isn't truly enough information to tell you about a person's chances of acceptance. You shouldn't listen to this light-hearted blog post or me at all for serious advice about college.

A Mix of Colleges: 2 safeties, 4 targets, and 2 reaches

The College Board and some other organizations recommend that half of the colleges you apply to are in the "target" category, and the other half is split between the "safety" and "reach" categories. For this section and the rest of this post, I will use eight colleges in the examples. The binomial can't be used here because of this mix of acceptance rates.

P(rejected from all 2 safeties, 4 targets, and 2 reaches)

= (1 - 0.875)^2 * (1 - 0.55)^4 * (1 - 0.175)^2

= 0.04%

P(accepted into at least 1 of 2 safeties, 4 targets, or 2 reaches)

= 1 - ((1 - 0.875)^2 * (1 - 0.55)^4 * (1 - 0.175)^2)

= 99.96%

P(accepted into all 2 safeties, 4 targets, and 2 reaches)

= (0.875)^2 * (0.55)^4 * (0.175)^2

= 0.21%

Only Reaches

If you hate safeties and only want to apply to colleges that aren't easily within your range, then this is your section.

P(rejected from all 8 reaches)

= (1 - 0.175)^8

= 21.5%

P(accepted into at least 1 of 8 reaches)

= 1 - binomcdf(8, 0.175, 0)

= 78.5%

P(accepted into at least 2 of 8 reaches)

= 1 - binomcdf(8, 0.175, 1)

= 42.1%

P(accepted into at least 3 of 8 reaches)

= 1 - binomcdf(8, 0.175, 2)

= 15.1%

P(accepted into at least 4 of 8 reaches)

= 1 - binomcdf(8, 0.175, 3)

= 3.61%

P(accepted into at least 5 of 8 reaches)

= 1 - binomcdf(8, 0.175, 4)

= 0.57%

P(accepted into at least 6 of 8 reaches)

= 1 - binomcdf(8, 0.175, 5)

= 0.06%

P(accepted into at least 7 of 8 reaches)

= 1 - binomcdf(8, 0.175, 6)

= 0.003%

P(accepted into all 8 reaches)

= 0.175^8

= 0.00009%

Only Ivy League

The Ivy League is a group of schools that are notoriously difficult to be accepted into. The average acceptance rate of an Ivy League school is extremely low: around 4.5%.

P(rejected from all 8 Ivy League universities)

= (1 - 0.045)^8

= 69.2%

P(accepted into at least 1 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 0)

= 30.8%

P(accepted into at least 2 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 1)

= 4.73%

P(accepted into at least 3 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 2)

= 0.43%

P(accepted into at least 4 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 3)

= 0.02%

P(accepted into at least 5 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 4)

= 0.0009%

P(accepted into at least 6 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 5)

= 0.00002%

P(accepted into at least 7 of 8 Ivy League universities)

= 1 - binomcdf(8, 0.045, 6)

= 0.0000003%

P(accepted into all 8 Ivy League universities)

= 0.045^8

= 0.000000002%

Only UCs

The University of California system offers nine undergraduate universities at various acceptance rates. Unfortunately, the binomial can't be used here either because of this. I will use the acceptance rates of each school provided by College Vine.

P(rejected from all 9 UCs)

= (1 - 0.89) * (1 - 0.69) * (1 - 0.47) * (1 - 0.37) * (1 - 0.26) * (1 - 0.24) * (1 - 0.21) * (1 - 0.11) * (1 - 0.09)

= 0.41%

P(accepted into at least 1 of 9 UCs)

= 1 - ((1 - 0.89) * (1 - 0.69) * (1 - 0.47) * (1 - 0.37) * (1 - 0.26) * (1 - 0.24) * (1 - 0.21) * (1 - 0.11) * (1 - 0.09))

= 99.6%

P(accepted into all 9 UCs)

= 0.89 * 0.69 * 0.47 * 0.37 * 0.26 * 0.24 * 0.21 * 0.11 * 0.09

= 0.001%

Only the top four UCs (by acceptance rate)

Often, some students only apply to the UCs that are the most difficult to be accepted into. The top half of the UCs includes Los Angeles, Berkeley, Irvine, and San Diego.

P(rejected from all of the top four UCs)

= (1 - 0.24) * (1 - 0.21) * (1 - 0.11) * (1 - 0.09)

= 48.6%

P(accepted into at least 1 of the top four UCs)

= 1 - ((1 - 0.24) * (1 - 0.21) * (1 - 0.11) * (1 - 0.09))

= 51.4%

P(accepted into all of the top four UCs)

= 0.24 * 0.21 * 0.11 * 0.09

= 0.05%

Only Safeties

What if you want to guarantee your admission into any college, no matter the acceptance rate?

P(rejected from all 8 safeties)

= (1 - 0.875)^8

= 0.000006%

P(accepted into at least 1 of 8 safeties)

= 1 - binomcdf(8, 0.875, 0)

= 99.99999%

P(accepted into at least 2 of 8 safeties)

= 1 - binomcdf(8, 0.875, 1)

= 99.9997%

P(accepted into at least 3 of 8 safeties)

= 1 - binomcdf(8, 0.875, 2)

= 99.99%

P(accepted into at least 4 of 8 safeties)

= 1 - binomcdf(8, 0.875, 3)

= 99.9%

P(accepted into at least 5 of 8 safeties)

= 1 - binomcdf(8, 0.875, 4)

= 98.9%

P(accepted into at least 6 of 8 safeties)

= 1 - binomcdf(8, 0.875, 5)

= 93.3%

P(accepted into at least 7 of 8 safeties)

= 1 - binomcdf(8, 0.875, 6)

= 73.6%

P(accepted into all 8 safeties)

= 0.875^8

= 34.4%

Only Targets

Maybe there are a few schools that you want to go to that you believe are right within your reach.

P(rejected from all 8 targets)

= (1 - 0.55)^8

= 0.17%

P(accepted into at least 1 of 8 targets)

= 1 - binomcdf(8, 0.55, 0)

= 99.83%

P(accepted into at least 2 of 8 targets)

= 1 - binomcdf(8, 0.55, 1)

= 98.2%

P(accepted into at least 3 of 8 targets)

= 1 - binomcdf(8, 0.55, 2)

= 91.2%

P(accepted into at least 4 of 8 targets)

= 1 - binomcdf(8, 0.55, 3)

= 74.0%

P(accepted into at least 5 of 8 targets)

= 1 - binomcdf(8, 0.55, 4)

= 47.7%

P(accepted into at least 6 of 8 targets)

= 1 - binomcdf(8, 0.55, 5)

= 22.0%

P(accepted into at least 7 of 8 targets)

= 1 - binomcdf(8, 0.55, 6)

= 6.32%

P(accepted into all 8 targets)

= 0.55^8

= 0.84%

Personalized Chances

Not every college has the exact acceptance rates of 17.5%, 55%, or 87.5%. For more personalized chances, you should use my Python program. Enter the acceptance rates of each college that you hope to be admitted into, and it will give you the probability of being accepted into all, one, or none of the colleges.

College probabilities

You should also check out Plannter, my website that lets you plan courses, keep track of extracurriculars, prepare for college applications, and more.