Marble Bag Pull

Solution

We will calculate and compare the expected value of pulling out a marble and the current balance. At the beginning, the bank balance is $0. The expected value of pulling out a marble is given by: \[ \frac{13}{20} * $5 + \frac{7}{20} * $0 = $3.25 \] Since \( $3.25 \) is greater than the bank balance (\( $0 )\), it is profitable to pull out a marble.

Now, we have a balance of \( $5 \). The expected value of pulling out another marble is given by: \[ \frac{12}{19} * $10 + \frac{7}{19} * $0 \approx $6.32 \] Once again, since \( \approx $6.32 \) is greater than the bank balance (\( $5 )\), it is profitable to pull out a marble.

The expected value of pulling out another marble is given by; \[ \frac{11}{18} * $15 + \frac{7}{18} * $0 \approx $9.17 \] Since \( $9.17 \) is less than the bank balance (\( $10 )\), it is not profitable to pull out a marble.

Therefore, the optimal strategy is to pull out a maximum of 2 marbles and then quit.