# Guessing answers on a test

by mitjak
Tags: false, probability, test, true
P: 8
I've been struggling with this problem for two days. It is out of the Concepts in Probability and Stochastic Modelling textbook (1.4-13), and it goes like this:
 A true false exam has 15 questions of which 5 are true and 10 are false. A student randomly selects seven questions and answers those false. The remainder of the questions the student answers true. The best possible score the student could obtain on this exam is 12 correct answers. What is the probability of this happening?
The problem appears in one of the introductory chapters, in this case the one on combinations and permutations, but to me it seems like the problem is a bit more complex than that. For instance, when choosing the first 7 questions, the student is guessing, so the odds of him picking the right answer are:
$$({\frac{2}{3}})^7$$.

The sample space doesn't change as he goes about guessing the answers, so neither combinations nor permutations apply here, it seems. But then, when he fills out the remaining 8 true answers, he's no longer randomly picking answers but just marking those remaining 8 answers as false.

I am probably overthinking the problem, but at this point I'm stuck. Would greatly appreciate any help.

EDIT: crap. Didn't realize I'm not supposed to post in this forum. Apologies. Reposting in the coursework forum.

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