This is a question from a homework assignment I am working on. I am using numbers in this example, although the question can and should be proven in the general case for n, m and k. New with mathjax so doing my best here.
There are $n = 1000$ cards, $m = 50$ cards are blue and $n - m = 950$ cards are red. We randomly draw, without replacement, $k = 25$ cards. Prove that the probability we draw no blue cards is at most $(\frac{(n-k)}{n})^m$.
My approach so far is using the fact that the probability of drawing no blue cards == the probability that the first card drawn is red, times the probability that the 2nd card is red, given the first card is red, times the probability that the 3rd card is red, given the first 2 cards were red, ..., times the probability that the $k^{th}$ card is red, given the first k-1 cards were red. I can clean up this product of probabilities using factorials, but am having trouble comparing it to the objective upper-bound for the proof.
Appreciate any thoughts on this.