# Calculating the probability of 31 of 628 items are sampled (no replacement) more than 10x amongst 150 participants drawing 50 items each

I am sorry for having to ask a simple probability question, but I have been thinking about it for weeks and an extensive google search has given no answers.

I have a group of 628 questions. 31 Questions belong to category A. 150 participants will each be given 50 randomly sampled questions from the total 628 questions (no replacement). What is the probability that questions belonging to category A will be answered more than 10 times by the group of participants?

I looked at a binomial distribution using R's binomial density function, but the best I could come up with was the probability of 1 item from category A being administered to the 150 participants if only 1 question was sampled from the total of 628 questions.

    x <- seq(1,50,by=1)
high.biDen <-dbinom(x,size=150,prob=((31/628)))
round(high.biDen,2)

round(sum(high.biDen[10:50]),2)


The answer obtained is .208 or a 20.8% probability that a question from category A will be administered to the participants more than 10 times if 1 question is sampled from the pool of 628 questions. I would like to know the probability if 50 questions are sampled instead of 1.

Is the answer obtained by some sort of bayesian technique? It seems to rely heavily on conditional probability seeing how if one item is sampled only 30 remain to be randomly selected.

Thanks so much for sharing your knowledge and expertise!

-Xander

• Please share your calculations thus far in the question. It's hard to help you until you give more disclosure on exactly what you've done.
– John
Nov 5, 2013 at 5:02

## 1 Answer

(1) What is the probability $p_m$ that one participant receives exactly $m$ A-questions? For $m>31$ it is zero, otherwise

$$p_m = { {31 \choose m} {597 \choose 50-m} \over {628 \choose 50} }$$

(Explanation: we should select $m$ A-questions than $(50-m)$ non-A-questions, and there are ${31 \choose m} {597 \choose 50-m}$ ways to do it.)

(2) What is the probability $q_{k,m}$ that $k$ participants together receive exactly $m$ A-questions? For k>0:

$$q_{k,m} = \cases {p_m & if k=1 \\ \sum_{n=0}^{\min(m,31)} p_n q_{k-1,m-n} & otherwise}$$

(Explanation: let $n$ be the number of A-questions received by the last participant. The probability that he/she received $n$ A-questions, and previous participants received $m-n$ questions, is $p_n q_{k-1,m-n}$. Since we don't know how A-questions he receive, we should sum it by $n$.)

• Hello, @user31264, thank you for your response. It looks like a truly elegant answer. I can tell that you are very well versed in theoretical statistics and the respective equations. As a social scientist who is simply a statistics user. How would I be able to solve these equations if m = 10? Is there a resource you could direct me towards to see about solving this to get a final percentage? Thank you for sharing your time and expertise. Nov 5, 2013 at 20:13
• Hello, the simplest way is to write an R program with two loops. Nov 5, 2013 at 20:43