Gambler's ruin - problem 6

You walk into Martin Gale's Betting Room with an initial budget of \(k\)

As usual, you can play the following game any number of times (if you still have what it costs)

  • You pay Martin a dollar.
  • Martin tosses a fair coin
  • If the coin comes up heads Martin pays you two dollars. If it comes up tails, you get nothing.

Your plan is to just keep playing indefinitely, if you can.

Which of the following is true?

1) You will lose all your money within \(O(k^2)\) games.

2) You will lose all your money within \(O(2^k)\) games.

3) There is an integer \(n = n(k)\) such that it is impossible to increase your wealth to \(n\)

4) For each integer \( n > k\), you have a positive probability of increasing your wealth to \(n\), hence you have a positive probability of being able to play indefinitely.

5) For each integer \( n > k\), you have a positive probability of increasing your wealth to \(n\), but you will inevitably eventually lose all your money.

Note: You may assume that Martin's budget is infinite. In fact this is implicit in the fact that you are even allowed, in principle, to play indefinitely. If Martin had a finite budget \(B\) then situation would be equivalent to your deciding to play up to threshold \(k+B\).

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