Ms. French, a bank robber, robs banks by entering the bank at night when the bank is empty and then entering the bank vault. She is considering robbing three different banks. She will only rob bank 2 if she successfully robs bank 1, and she will only rob bank 3 if she successfully robs bank 2. She can, however, choose not to rob a bank even if she was successful in robbing the preceding bank. If the banks are successfully robbed, she expects to make $10 thousand, $50 thousand, and $75 thousand from banks 1,2, and 3, respectively.
To successfully rob a bank, two things must occur. First, Ms. French must enter the bank without setting off the outside alarms and then, she must enter the vault without setting off the vault alarms. The probability of entering the first bank without setting off the outside alarms is 0.6, and the probability of entering the vault without setting off the vault alarms is 0.25. For bank 2 those probabilities are 0.4 and 0.2, and for bank 3 the probabilities are 0.35 and 0.2. If Ms. French fails in entering the bank, she is capable of escaping before the police arrive, thus she will not lose anything in such a scenario. If she sets off a vault alarm she will be arrested and incur a penalty of $30,000.
Ms. French can hire a sidekick before any of the robberies, but once the sidekick is hired, he will continue to be used for any and all subsequent robberies. The cost of the sidekick is $2500 for bank 1, $12,500 for bank 2, and $18,750 for bank 3. This amount must be paid whether or not the robbery is successful.
If the sidekick is hired, the probability of gaining entry to the vaults at banks 1, 2, and 3 improves to 0.3, 0.25, and 0.25, respectively. The effect of the sidekick on entering the bank is unclear, but we do know that the conditional probability of having a sidekick given that bank 1 was successfully entered is 0.6. For bank 2 this probability was 0.75, and for bank 3 this probability was 0.8.
Develop a decision tree for this problem and determine Ms. French's optimal strategy (which banks to rob and when, if ever, to hire a sidekick) if she wishes to maximize her expected reward.