Suppose there is a safe which automatically keeps track of how many \$1 bills are inside of it. This safe is shared between two people. Whenever one of the people accesses the safe, they take out a \$1 bill. The safe is slow meaning that the balance displayed on the safe's digital display is only refreshed every X minutes. When the safe is finally emptied of all the money by one of the two people, there is a period of time P (0 < P < X) during which the other person doesn't yet know and may attempt to try to withdraw a bill despite it being empty. The two people withdraw the \$1 bills at different independent average rates (U1 and U2 bills per minute) and they won't attempt to withdraw a bill if they know the safe is empty. If the safe currently contains Y \$1 bills, then what's the probability that, in the next D minutes, one of the people will unknowingly attempt to withdraw a bill while it is empty?
2026-04-06 03:13:24.1775445204