Your bank starts at 0 dollars. You may press a button at most 5 times. Each press, independently, adds 4 dollars with probability 3/4, and with probability 1/4 wipes the bank out and ends the game with 0. Before each press you may stop and keep the bank. With optimal play, what is the expected final bank?

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Growing bank risk

Expected valueHard

Your bank starts at $0$ dollars. You may press a button at most $5$ times. Each press, independently, adds $4$ dollars with probability $\frac{3}{4}$ , and with probability $\frac{1}{4}$ wipes the bank out and ends the game with $0$ . Before each press you may stop and keep the bank. With optimal play, what is the expected final bank?

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Hint

Find the bank size at which one more press stops being worth it.

Solution

From a bank of $b$ one more press is worth $\frac{3}{4} \times (b + 4)$ , which is above $b$ exactly when $b < 12$ .
So you press until the bank reaches $12$ (at $12$ you are indifferent) and then stop.
Reaching $12$ needs $3$ successful presses in a row, which fits within $5$ presses and has probability $(\frac{3}{4})^{3} = \frac{27}{64}$ .
The expected final bank is $\frac{27}{64} \times 12 = \frac{81}{16}$ .

Answer: $\frac{81}{16}$

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