Hilbert's Casino (pt. 1)

Calculus Level 2

Your friend, David Hilbert, runs a casino with space for an infinite number of guests. He installs a slot machine with the following properties:

  • Each guest at the casino plays one round at the slot machine after all the guests have arrived.
  • If there are \(n\) guests at the casino, then the odds of winning the slot machine are always \(\frac{1}{n}\) for all of them.
  • When a guest wins, they get all of the money in the machine.

To understand how profitable this machine is, he wants to know the probability that no one wins the slot machine.

If David fills up his casino with a number of guests approaching infinity, what does this probability approach?

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