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If the probability of me getting into Dartmouth is 0.3 and the probability of me getting into Cornell is 0.3, what is the probability that I will get into at least one of the schools?

I believe the answer is simply .3 + .3 - (.3 * .3) = .51

However, my professor is saying these two events aren't independent and therefore we can't answer the question.

I'm not clear on why these aren't independent. Of course each school is looking for similar things, but the acceptance of one school does NOT affect the acceptance of another. Therefore they are independent! (That's my thought process)

Where am I incorrect? The professor is a harvard PHD so I'm assuming I'm wrong but I need to understand why.

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    $\begingroup$ In a recent LinkedIn post, John Byrne quotes an "admissions consultant" who claimed admissions officers at Stanford's business school screen applicants who would be suited for Harvard's business school--and reject them on that basis! Although this might be pure speculation, it provides reasons why an assumption of independence in this context is not only unwarranted, but could be grievously in error. If both Dartmouth and Cornell used this approach, your chance to get into at least one might be zero! $\endgroup$
    – whuber ♦
    Feb 4, 2015 at 22:37
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    $\begingroup$ There's not enough information given to answer the question. Whatever the answer is, it must be no more than 0.6, but could be much lower than 0.6. $\endgroup$
    – Glen_b
    Feb 4, 2015 at 23:13
  • $\begingroup$ In addition to clarifying your question, you need to add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. $\endgroup$
    – gung - Reinstate Monica
    Feb 4, 2015 at 23:18
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    $\begingroup$ To clarify my comment: Consider that "getting in" is not just "random" (it's not like the admissions people draw names out of a hat or something) -- acceptance depends heavily on the characteristics of the student. There's typically likely to be strong positive dependence (since they like some of the same characteristics about students, if you get into one, there's a good chance of getting into the other; and since they also tend to dislike the same things about applicants, if one rejects you, the other is more likely to as well). The answer might be much nearer 0.3 than independence implies. $\endgroup$
    – Glen_b
    Feb 4, 2015 at 23:24
  • $\begingroup$ Re the edited question: the comments at stats.stackexchange.com/questions/116355 and the answers at stats.stackexchange.com/questions/3869 might help you better appreciate what "independent" means. $\endgroup$
    – whuber ♦
    Feb 9, 2015 at 19:11

2 Answers 2

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Let $A$ = the event of getting into Dartmouth.

Let $B$ = the event of getting into Cornell

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) = P(A) + P(B) - P(A|B)P(B) = P(A) + P(B) - P(B|A)P(A)$

Assuming independence, $P(B|A) = P(B)$ so we get $P(A \text{ or } B) = .3 + .3 - .3(.3) = .51$. If not independent, we need more information. Namely, either $P(A|B)$ or $P(B|A)$.

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  • $\begingroup$ This is what I thought it was (.51) - but my professor (i'm an MBA student at Cornell right now) said that because the two events are related we can't answer the question. Im assuming thats what you mean by "assuming independence" - but I don't understand why they ARENT independent...any ideas? $\endgroup$
    – Tyler hogge
    Feb 5, 2015 at 3:47
  • $\begingroup$ Tyler, that question was answered by @Glen_b in his latest comment and in my initial comment. Note, please, that nobody is asserting those two events are necessarily independent. We are only saying (1) you don't have enough information to assume they are independent; (2) there are reasons to suppose that assumption would be invalid; and (3) its invalidity can greatly change the result. $\endgroup$
    – whuber ♦
    Feb 9, 2015 at 17:01
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The probability is the sum of your chance to going to Cornell plus the chance of not going to Cornell times the chance of going to Dartmouth - assuming these chances are independent. Thus: $p=0.3+0.7 \times 0.3 = 0.51$

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  • $\begingroup$ You still can get into Dartmouth AND Cornell, it is implied in the first term "0.3" (precisely, you get $0.3 \times 0.3$ chances for this case), but you don't care since what you want is to compute the probability to get accepted "at least" in one of these school. $\endgroup$
    – gdupont
    Feb 4, 2015 at 22:19

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