How Often to Throw the Balloon
You and your partner each hold one water balloon, caught with probability 1/2 when thrown, independently of everything else. Before the round you each decide, without communicating, whether to throw. You win if at least one balloon is thrown and every thrown balloon is caught; anything else loses.
You agree beforehand on one rule and you both follow it: throw with probability , independently of each other.
Which maximizes the probability of winning? Answer format: round to 3 decimal places.
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