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From what I understand (which is not much), private key is like id/password for your wallet. So lets say, I have random number generator and I generate random number, is it possible that generated number could be key to someone elses wallet?

Jay
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Yes, it is possibly, but not probable, or in other terms, extremely rare. There'll be maximum 2^160 addresses, which is much more than the number of addresses that are used. See this.

MCCCS
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    The language really needs other words for this level of probability. I mean we use those words to describe the odds of finding an almost extinct frog in the jungle. The odds of that occuring is indescribable higher than the odds of guessing a random private key. – Christiaan Westerbeek Dec 13 '17 at 16:58
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    It's about the chance of winning the Powerball jackpot 6 years in a row, with just one ticket. – Pieter Wuille Dec 13 '17 at 18:19
  • Bon appetit @Jay, enter your random generated number here and check if you are luckier than the Powerball winner mentioned above. – Chak Dec 13 '17 at 18:33