the intuitive difference between expected utility and utility of expected profit in a gambling game

Question

What is the intuitive difference between expected utility and utility of expected profit in a gambling game ? Which one is the "usefulness of the game" to a player ?

Answer 1

Example:

There is a game with two possible results $P_1=100, P_2=0$. The probability for each result is $p=1-p=0.5$ The Player have the utility function $U(P_i)=\sqrt{P_i}$

$\text{Expected Utility}= E(U(P)) =0.5 \cdot \sqrt{ 100} +0.5\cdot \sqrt 0=5$

$\text{Utility of the Expected Value}=U(E(P))=\sqrt{0.5 \cdot 100+0.5 \cdot 0}=\sqrt{50}\approx7.071$