There’s definitely bet patterns superior to risking everything—but only if you’re allowed to play a very large number of rounds
If the pharmacist is only willing to play 10 rounds, then there’s no way to beat betting everything every round
As the number of rounds you play approaches infinity (assuming you can bet fractions of a coin), your chance of saving everyone approaches 1. But this takes a huge number of rounds. Even at 10,000 rounds, the best possible strategy only gives each person around a 1 in 20 chance of survival
There’s definitely bet patterns superior to risking everything—but only if you’re allowed to play a very large number of rounds
If the pharmacist is only willing to play 10 rounds, then there’s no way to beat betting everything every round
As the number of rounds you play approaches infinity (assuming you can bet fractions of a coin), your chance of saving everyone approaches 1. But this takes a huge number of rounds. Even at 10,000 rounds, the best possible strategy only gives each person around a 1 in 20 chance of survival