Hopefully my title isn't misleading... i'm not sure what this math problem is called, making searching for the answer quite difficult. Maybe someone can tag this question appropriately if you know the name for what i'm asking.
The Question:
Let's suppose I am applying to get into a variety of schools, each one with a different acceptance percentage (defined as the number of students they accept out of the number of applications they receive). How do I determine what the likelihood is that I would be accepted into at least one of the schools, given I know the acceptance rate of each?
Example/Data:
School A: 50% Acceptance Rate
School B: 25% Acceptance Rate
School C: 25% Acceptance Rate
School D: 20% Acceptance Rate
Given the above data, what is the likelihood I'll be accepted into any (at least one) of schools A, B, C, or D.
What I know (or think i know)
- It has to be at least as high as the highest percent (using the example, 50%)
- If they were all 50% (like a coin-toss), the chances of getting all declined is 6.25% (1 in 16), leaving the remaining 15 possibilities, or 93.75%, with me getting into one or more schools.
- No matter how many schools I add, the answer can never be equal to, or over 100%, unless one of the school's acceptance rates is 100% (asymptote?)
My own attempt at an answer
I started plugging numbers into excel based on some gut feelings on how to approach this. I used the coin-flip logic to test my calculations, then applied it to the example data. I ended up with answer of 77.5%, but am not sure if this is correct.
Here is a screenshot of my function and table:
(Note cell E4 with the same 93.75%)
Closing Remarks
This question was sparked in my office in response to this website which shows an "Aggregate Admission Probability" in which they simply add the percents. This seemed completely wrong and useless to me. Thought I would turn to the internets to settle the dispute :)
Hopefully this isn't something really easy that I am thinking is more complicated than it is. If that's true, let me down gently ;)
I assume the probabilities are independent, meaning that one school's decision has no influence on the result of another. This is probably false, but it's the only way a definitive answer can be given (unless you know exactly what the dependence is).
The probability of getting into at least one school is $1$ minus the probability of getting rejected from every school. Assuming independence, this is just the product of the probabilities of getting rejected from each school. In your example, we get $$1-(1-.5)(1-.25)(1-.25)(1-.5)=.859375=85.9375\%$$ as the probability of getting accepted into at least one school.