The Gumball Proposition
Suppose there are two gumball machines outside a grocery store, each initially containing \(100\) gumballs. Each machine has an equal chance of being chosen by people wanting a gumball, with each machine dispensing one gumball at a time.
Now the gumball containers are such that no one can tell how many gumballs are left in a machine until someone tries to buy a gumball only to find that that machine is empty. On the first occasion that someone discovers that one of the machines is empty, the probability that there are (strictly) more than \(5\) gumballs left in the other machine is \(S.\)
Find \(\lfloor 10000*S \rfloor.\)