Correct solutions came from Philippe Fondanaiche, France; Juan Carlos Marivela, Spain.

It takes at least six drawings to have a 50% chance of having (at least) five black balls.

The only way to have five black balls after five drawings is to have drawn BBBBB which occurs with probability .3024; not enough.  After six drawings we can have, additionally, BBBBWB which occurs with probability .12096, or BBBWBB which occurs with probability .09072, and two cases.  The sum of these three probabilities is already above .5.