May someone help me with this probability problem, please?
In a lottery game, one random number between N numbers is choosen. One player chooses $n<N$ numbers in a single game. Other player chooses just one number for each $n$ games he plays. Question: Which player has the higher probability of winning the game?
Now what I know. First player probability of winning is : $\frac{n}{N}$
Probability of second player loses every game: ($1-\frac{1}{N})^n$
Probability of second player wins at least one game: $1-[(1-\frac{1}{N})^n]$
So, now I only need to prove that $\frac{n}{N}$ is higher or lower than $1-[(1-\frac{1}{N})^n]$ knowing that $N>n>1$, since for $n=1$ the probabilities are equivalent. I've tried and tried, but no success.
Also, I'm open to different solutions, but I'm in my third week of college, so I may understand only high school math.
Lastly, this is my first post in this forum and English is not my first language, so if there's something I did wrong, please tell me so I can change it.