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The Prince and the Pearls ★★

Once upon a time there was a young Chinese prince. He wanted to marry the daughter of a mandarin. The mandarin decided to test the prince. He gave the prince two empty porcelain vases, 100 white pearls, and 100 black pearls. "You can distribute any pearls on the vases," he said to the prince. "Afterwards I'll call my daughter. She will pull any pearl from one of these two vases. If this pearl is black, you can marry my daughter."

The question: How did the prince have to distribute the pearls so that his chances were as high as possible?

Another question: You have three vases: a vase with two white pearls, a vase with one white and one black pearl, and a vase with two black pearls. A pearl is pulled from one of these vases. It turns out that this pearl is white. What is the chance that the other pearl in the same vase is also white?

And another question: You have ten vases: five of these vases contain a white pearl and four of these vases contain a black pearl (note that a vase can contain both a white and a black pearl!). You can choose any of these vases. What is the chance that the vase you choose is empty?