$32$ points

Suppose that a simple environment has $3$ states which are encoded as integers $0,1$ and $2.$ The environment has two actions, $0$ and $1.$

The dictionary below provides the transition probabilities when taken action $1$ from state $0.$ For example, this dictionary tells use that $p(s^{\text{'}}=2|s=0,a=1)=0\mathrm{.}3$

$\{0$:$0\mathrm{.}2,1$:$0\mathrm{.}5,2$:$0\mathrm{.}3\}$

The dictionary below explains the rewards earned by the agent when transitioning to a particular state from state $0.$

$\{0$:$0,1$:$4,2$:$2\}$

Suppose that the state$-$value function $V_{}(s)$ for a certain policy $$ is represented by the dictionary below.

$\{0$:$18,1$:$13,2$:$15\}$

Use the Bellman equation and the information above to calculate $Q_{}(0,1)$ for the given policy. Assume a discount rate of $=0\mathrm{.}9.$

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