Fortunately, as the "creator" I am not obliged to maintain precisely the same scenario to get at the point so, let me suggest a slightly different scenario:
I have an enormous barrel in a pitch black room and I know that in the barrel there are two million balls, only two of which are entirely white, the rest being of various colours, shades and and patterns. I take my urn into the pitch black room and, completely at random, I take out two balls from the barrel and place them in the urn. Because it is so dark in the pitch black room, I cannot see either of the balls when I do this.
Then, while still in the pitch black room, I reach into the urn and, completely at random, I draw out one ball. Because it is so dark in the pitch black room, I cannot see either of the balls when I do this.
Finally I walk out of the pitch black room and I look at the ball I drew out of the urn. It is white.
What is the probability that the second ball - the one still in the urn - is white?