Sampling until first occurence

Question

Assume we have a bag of red and black marbles and we wish to sample from the bag without replacement until we find the first occurrence of a red marble (then we stop). If I am interested in finding the probability that no red marbles will be found amongst the first n marbles, does the probability change if I stop as soon as a red marble is found? In other words, is this problem equivalent to: the hypergeometric sampling of n marbles from the population, i.e. the probability that all n marbles are black in a sample of size n without replacement