Why I am getting below answer ?

Transition to state s′s's′ with probability 0.4, reward R′=10R' = 10R′=10, and v(s′)=0v(s') = 0v(s′)=0 (assumed).

Transition to state s′′s''s′′ with probability 0.6, reward R′′=3R'' = 3R′′=3, and v(s′′)=0v(s'') = 0v(s′′)=0 (assumed).

Discount factor γ=0.5\gamma = 0.5γ=0.5.

Formula:

v(s)=Probability to s′×(R′+γv(s′))+Probability to s′′×(R′′+γv(s′′))v(s) = \text{Probability to } s' \times \left(R' + \gamma v(s')\right) + \text{Probability to } s'' \times \left(R'' + \gamma v(s'')\right)v(s)=Probability to s′×(R′+γv(s′))+Probability to s′′×(R′′+γv(s′′))

Substitute the values:

v(s)=0.4×(10+0.5×0)+0.6×(3+0.5×0)v(s) = 0.4 \times \left(10 + 0.5 \times 0\right) + 0.6 \times \left(3 + 0.5 \times 0\right)v(s)=0.4×(10+0.5×0)+0.6×(3+0.5×0)

Simplify:

v(s)=0.4×(10+0)+0.6×(3+0)v(s) = 0.4 \times (10 + 0) + 0.6 \times (3 + 0)v(s)=0.4×(10+0)+0.6×(3+0) v(s)=0.4×10+0.6×3v(s) = 0.4 \times 10 + 0.6 \times 3v(s)=0.4×10+0.6×3 v(s)=4+1.8v(s) = 4 + 1.8v(s)=4+1.8 v(s)=5.8v(s) = 5.8v(s)=5.8

Thus, the value of state sss is 5.8.

4o mini