Climbing Back to Even

A market-making bot plays a sequence of rounds. Each round, independently, it books a win with probability 0.7 0.7 or a loss with probability 0.3 0.3 . It is programmed to shut down the first time its total wins equal its total losses. Given that the very first round is a loss, what is the expected total number of rounds the bot plays before shutting down?

Answer

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