# 222 Black and 333 white balls [duplicate]

You have a bag with 222 black balls and 333 white balls in it.
You also have an almost infinite supply of black and white balls that are not in the bag.

You do the following over and over again:

• Choose two balls uniformly at random from the bag, and remove them from the bag.
• If the balls were the same colour, put a black ball back in the bag.
• If the balls were opposing colours, put a white ball back in the bag.

You do this until there is only one ball left in the bag.

What is the probability that that ball is black?