In my AP Government class in preparation of a quiz we are given an unlimited amount of attempts at a practice quiz. There is a bank of 40 questions. Each practice quiz consists of 10 random questions (no repeats) drawn from this question bank. The real quiz is just like a practice quiz and takes 10 random questions (again, no repeats) from the bank. If I take, say, 14 practice quizes, what is the probability that on the 15th real quiz I would receive a question I have not seen before (one that has never been on a previous practice quiz)?
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$\begingroup$ What have you tried? Certainly you're also taking AP Statistics concurrently? $\endgroup$– Parcly TaxelCommented Feb 28, 2019 at 5:46
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$\begingroup$ @ParclyTaxel I am actually not taking AP stats. I asked my calc teacher and he gave some advice but no solution. I wrote a computer program that simulates this senario which allowed me to experimentally find the probability, which was about 17% taking 14 practice quizzes and about 2.5% taking 20 practice quizzes. Do these numbers seem reasonable? $\endgroup$– Jacob GCommented Feb 28, 2019 at 6:07
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$\begingroup$ Looks reasonable; see my answer. $\endgroup$– Parcly TaxelCommented Feb 28, 2019 at 7:14
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This answer is adapted from the first part of this other answer.
Without loss of generality, fix the 10 real questions that appear on the real quiz. The probability that $j$ specific real questions have not been seen after $k$ practices ($1\le j\le10$) is $$S_j=\binom{10}j\left(\frac{\binom{40-j}{10}}{\binom{40}{10}}\right)^k$$ Then by inclusion/exclusion the probability that at least one real question has been seen is $$\sum_{j=1}^{10}(-1)^{j+1}S_j$$ For $k=14$ the probability is $0.165970\dots$, and for $k=20$ it is $0.031333\dots$
answered Feb 28, 2019 at 7:14