previously you give the answer as below for same question so which one is correct?

The correct answer depends on the assumptions you are making about the values of the states (s') and (s'').

Two Scenarios

1. Scenario 1: Given values of (v(s') = 5) and (v(s'') = 3):

• If you use the values (v(s') = 5) and (v(s'') = 3) as initially stated in your problem description: [ v(s) = 0.4 \times (10 + 0.5 \times 5) + 0.6 \times (3 + 0.5 \times 3) ]
• This simplifies to: [ v(s) = 0.4 \times 12.5 + 0.6 \times 4.5 = 5 + 2.7 = 7.1. ]
• Answer: (\boldsymbol{7.1}).

2. Scenario 2: Assumed values of (v(s') = 0) and (v(s'') = 0):

• If you assume (v(s') = 0) and (v(s'') = 0) as stated in your second example: [ v(s) = 0.4 \times (10 + 0.5 \times 0) + 0.6 \times (3 + 0.5 \times 0) ]
• This simplifies to: [ v(s) = 0.4 \times 10 + 0.6 \times 3 = 4 + 1.8 = 5.8. ]
• Answer: (\boldsymbol{5.8}).

Conclusion

• If (v(s') = 5) and (v(s'') = 3) are the given or known values for the states (s') and (s''), then the correct answer is 7.1.
• If you assume (v(s') = 0) and (v(s'') = 0) (as in a situation where these are initial values or for some hypothetical reasoning), then the correct answer is 5.8.

You should use the scenario that matches your assumptions or the context provided in the problem. If the problem explicitly gives you the values (v(s') = 5) and (v(s'') = 3), then 7.1 is the correct answer. If not, and you're assuming both are zero, then 5.8 is correct.

Please confirm which assumptions apply to your situation to choose the appropriate answer.