here's a small question I asked myself and I would be happy to discuss it with you.
2 friends (Oscar and Jake) are organizing movie nights. Each of the 2 participant choose $n$ movies. Every night a movie will be randomly choosen and watched. every movie can only be choosen once. What is the probability that the first $n$ movies are all Jake's (or Oscar's) movies ?
Here's my attempt to solve the problem :
A movie choosen by Jake will be denoted by $J$ and a movie choosen by Oscar will be denoted by $O$. I am interested in the probability of the event :
$$ A= JJJ...JJJOOO...OOO \cup OOO...OOOJJJ...JJJ. $$
Using equiprobability, I guess the probability to be :
$$ P(A)=\frac{2}{\frac{(2n)!}{n!n!}}. $$
The 2 in the nominator comes from the fact that $A$ consists of 2 events. The denominator is the total number of permutation of the $2n$ movies. Is my reasoning correct ?
Also I have a second question. I just started teaching myself some probability and statistics and I would be really happy to solve this problem using random variables. Unfortunately I have no idea how to do it, is it even possible ? I guess I should use a binomial random variable, right ?
Thanks for your help.