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In my understanding, $\pi(a|s)$ is the probability that random policy $\pi$ chooses action a given the current state is $s$. And $\pi(a,s)$ is the probability that random policy $\pi$ chooses action a while the state is $s$. I think both of them mean the same and have the same probability. Is that true?

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As you said $\pi(a|s)$ is the probability that the policy chooses action in a given state s. But there is no such thing as $\pi(a,s)$. I believe by $\pi(a,s)$ you are talking about $q(a,s)$, which is actually the expected return when action a is taken at state s.

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