# Advanced Probability Question - Chance of Not Picking an Object with 10 Trys and no Replacement, then replacing all objects and repeating.

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)?

• What have you tried? Certainly you're also taking AP Statistics concurrently? Feb 28, 2019 at 5:46
• @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? Feb 28, 2019 at 6:07
• Looks reasonable; see my answer. Feb 28, 2019 at 7:14

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 $$0.031333\dots$$