You take care of n children and to keep them busy, you give to them (not to ever

Question

You take care of n children and to keep them busy, you give to them (not to every child individually) n toys, randomly choosing from a big box with n blue toys and n red toys. The children do not like red toys. If children get k red toys (for any k), they will return these k toys to you and complain. In this case, you will discard there k toys and you find another k toys chosen uniformly at random from the leftover toys in the big box. If they get red toys again and they may complain again, but at this point you just give up and do nothing. What is the probability that they get all the blue toys in the end?

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