Ruin

Probability Level 4

I start out with a loan of a million Jakedollars from the Bank of Jake. I then spend that money gambling, each time betting a fixed amount of money. If I win, I get what I bet; if I lose, I concede my bet.

If I go bankrupt, I get another loan. However, since my reputation is decreased, the second time I get a loan, I can only get a half a million Jakedollars instead. The third time has a cap of a third of a million Jakedollars, the fourth a quarter, et cetera.

What is the expected debt I owe to the bank - at the end of all time - in million-Jakedollars?

Note: It is possible to have a fraction of a Jakedollar. I win and lose 50% of the time each.

1/e

\ln 2

\sqrt{2}

\infty

Cannot be determined

\zeta (\frac{3}{2})

e

\ln^{2} 2

1 solution

Jake Lai
Jun 13, 2015

This problem is based on the concept of gambler's ruin . The idea is that, for a random walk in $\mathbb{Z}$ , I will land on every integer with probability $1$ . This means that, starting with any arbitrary amount of money, I will always return to $0$ .

Since I always need to get a loan from the bank, my debt is the sum of all my loans, ie

$1 + \frac{1}{2} + \frac{1}{3} + \ldots$

This is the well-known harmonic series, which diverges. Hence, I will be the most Jakebankrupt person in the multiverse, with a whopping debt of $\boxed{\infty}$ Jakedollars.

However, I'm safe because Jakedollars are completely worthless! (The proof is left as an exercise for the reader.)

Ruin

1 solution

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