Question about the risk analysis. 
In the above one can see the detail of this question, I am beginner in this kind of mathematics. I will be very greatful if any one can help me to solve them. 
 A: I'm assuming this is an assignment question so I'm just going to try to point you in the right direction (and only on the first part: you said in a comment that this was what you wanted help with).

Do any decision makers with (increasing) utility function agree about preferring risk $X_1$ to $X_2$?

The expectations of $X_1$ and $X_2$ are equal: $E(X_1)=E(X_2)$. So there would be no reason to prefer $X_1$ or $X_2$ on this basis.
The question is asking if there is any increasing utility function $u(x)$ such that
$$E(u(X_1)) > E(u(X_2)) \\ \iff \\
\sum_{x=0}^{10}u(x)Pr(X_1=x) > \sum_{x=0}^{15}u(x)Pr(X_2=x) \\ \iff \\
\sum_{x=0}^{10}u(x)\binom{10}{x}0.5^x0.5^{1-x} > \sum_{x=0}^{15}u(x)\binom{15}{x}(1/3)^x(2/3)^{1-x}$$
Note that $X_1$'s probability of success, 1/2, is greater than $X_2$'s probability of success, 1/3. But $X_2$ still has the same expectation as $X_1$ because with $X_2$ the maximum number of successes you can get, 15, is greater than the maximum number of successes you can get for $X_1$, 10. This suggests that if such a utility function exists, it is one which disproportionately rewards a lower number of successes.
The graph below shows the line $y=x$ in black for the utility function $u(x)=x$, corresponding to $E(u(X))=E(X)$. There are two other lines in green and red: if one of these corresponded to a utility function where decision makers would prefer $X_1$ to $X_2$, which one would it be?

