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This question already has an answer here:

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?

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marked as duplicate by Jaap Scherphuis, dcfyj, Community Oct 18 '17 at 12:26

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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At each step the number of white balls stays the same or decreases by 2. Therefore the parity of the number of white balls stays the same throughout. You start with an odd number of white balls, so when there is 1 ball left, you must still have an odd number of white balls. The last ball is therefore always a white one. The probability of it being black is 0.

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