How many topics to study to get 90% probability of covering something on the exam? I have a problem. Our professor is giving us at the exam the opportunity to choose to answer only one topic among eight on the exam. The total number of topics in the subject is twenty-two. 
I would like to know what is the minimum number of topics to study in order to get at least 90% probability to know at least one of the topics on the exam. I am assuming that all topics are selected with equal probability.
If it's possible I would like to have an explanation on the results.
 A: $^\textit{(The first version of this answer didn't account for the fact that topics are selected without replacement. Corrected.)}$
Assume the topics on the exam are chosen at random. Then (without loss of generality) assume you study the topics numbered $1, 2, ..., k$.
Then if the lowest numbered topic on the exam is $\leq k$ you will have studied at least one of the exam topics.
Let's work with the complement -- the probability that none of the topics we chose are on the exam. Imagine each topic in the subject is represented by a ball -- the $k$ topics we study are white balls and the ones we didn't ($22-k$) are black. The professor's random choice of topics selects $8$ balls at random from the $22$ available. What's the chance the professor draws only black balls?
The chance the first draw is black is $(1-\frac{k}{22}$. Given it was black the chance the second is black is $(1-\frac{k}{21}$, and so on down to $(1-\frac{k}{15}$. The probability all were black is the product, and the complement of that is the probability we're interested in:
So $\,\text{at least one white ball } = 1-(1-\frac{k}{22})\times(1-\frac{k}{21})\times...(1-\frac{k}{15})$ (we can write this as a product of combinations (it's just a hypergeometric) but I'll leave it in this form for now.
So (as long as the assumptions are reasonable) if you study $k$ topics this gives the probability there's at least one of those topics on the exam. 
    topics  %Chance >0
   studied   on exam
       1     36.36
       2     60.61
       3     76.36
       4     86.32
       5     92.40
       6     95.98
       7     97.99
       8     99.06
       9     99.60

(If the exam topics aren't chosen completely randomly with equal probability, you'd need to choose study topics at random to get these values)
A 10% chance (well, more like 8% as it turns out) of not having studied any of the topics seems like a bit of a risk but the required minimum number of topics is $5$, as we see in the table (not 6 as I said earlier) 
