History Study Guide Probability. The question I have is about a study guide I have to make. My history professor is giving us a study guide with 24 topics on it. He then chooses 10 of the 24 problems for the test. We are only required to choose 2 of those 10 problems to write about.
Now my question is how many of the 24 problems should I study to have a probability of less than 1% that I don't know both problems.
I know that 14/24 is the chance that one of the questions is not part of the 24 questions, however after this I don't know where t go. 
 A: Suppose you study $n$ questions.  The probability that you succeed is $1$ minus the probability that you fail.  You fail if the prof chooses all $10$ questions from the $24-n$ you didn't study or $9$ questions that you didn't study, and $1$ that you did.  Assuming the prof chooses the questions uniformly at random from the $24$, the probability of success is $$P(n)=1-\frac{\binom{24-n}{10}+\binom{24-n}{9}\binom{n}{1}}{\binom{24}{10}}$$ 
I get $$P(10)=0.9841779961412482\\P(11)=0.992198366760892,$$ so the answer to your question is $11.$  I doubt that the assumption that the professor chooses the questions uniformly at random is completely realistic, though.  Surely some questions will seem more important than others, and the prof is likely to be more interested in some topics than others.    
A: There are 2 ways to calculate the probability that you have studied at least 2 of the 10 questions if you have studied x problems.
The first is to treat your studying as fixed, and the 10 questions chosen by the professor as random.
$$ P(x) = 1 - \frac{\binom{x}{0}\binom{24-x}{10}}{\binom{24}{10}} - \frac{\binom{x}{1}\binom{24-x}{9}}{\binom{24}{10}} $$
The other is to treat the professor's chosen questions as fixed, and your studying as random.
$$ P(x) = 1 - \frac{\binom{10}{0}\binom{14}{x}}{\binom{24}{x}} - \frac{\binom{10}{1}\binom{14}{x-1}}{\binom{24}{x}} $$
Both methods tell you that you should study at least 11 questions to have at least a 99% probability.  However, I think you should focus on the 2nd.
It is very likely that the professor does NOT choose the 10 questions at random, he probably has unknown criteria that he uses, so those questions are FIXED.  What this means, is for the probability to be valid, you need to choose your 11 questions to study as a simple random sample.
You start by numbering all 24 questions, let's say 1-24.  Then you generate 11 random numbers from 1 to 24.  You could make small slips of paper also numbered 1-24 into a hat or whatever, then draw 11 of them to choose the 11 questions, or you could use a random number generator on a computer, or you could use a random number table.
What this means is if you draw a number associated with a question you HAVE to study it.  No matter how annoying the question is.  If you choose to skip a randomly chosen question, this probability will no longer be valid.  If you do want to skip questions because of them being annoying, then you will need to study 16 questions to be sure that you can pass the exam, because if you study 16 questions well enough, then you will be guaranteed to be prepared.
Therefore, the choice is up to you, choose 11 questions randomly and study them all no matter how annoying they are, or choose the 16 questions that you think are the easiest and study them.
