A simple stochastic game setup. There are two boxes. In one box there are two $100 bills, in the other one $100 bill and one fake bill worth nothing. The fake bill feels the same as a $100 bill and to the candidate it is indistinguishable from a real bill when picking it.
The person chooses a box and a bill in that box. The candidate is lucky and pulls our a $100 bill, which they get to keep.
Now the candidate is offered to do it again. What strategy should the candidate use to maximize their chances of pulling a second $100 bill? Should they switch to the other box or pull the second bill out of the first box? Following your strategy, what are the candidate's chances of winning another $100?
Recall the Monty Hall problem and consider that probability is not inherent to the world but merely a reflection of our state of knowledge. Particularly, after holding the $100 from one box in your hand, what are the odds that it was the box with two $100 bills?
It does not matter if we switch the box or if we stay.
If we stay on the box, then we must hope that the box we pulled the $100 the first time had two such bills. After observing the first $100 bill from that box, the chance that it is the box with two such bills has climbed from 50% to 67%.
On the other hand, if we switch, then we must either have taken the first $100 from the box with the worthless fake (chance 1/3) or we were were on the box with two $100 bills (chance 2/3) and we pull the real bill from the box with the fake.
In both cases, our odds of winning another $100 bill are 2/3.