= Solution
\b[Yes: the stated order has an <expected utility> representation.] For outcomes whose monetary amounts are $1,2,4$, respectively, choose
$$
u(1)=2,\qquad u(2)=0,\qquad u(4)=3.
$$
Then
$$
\begin{aligned}
U(\ell_1)&=\tfrac12\,0+\tfrac12\,3=\tfrac32,\\
U(\ell_2)&=\tfrac35\,2+\tfrac25\,3=\tfrac{12}{5},\\
U(\ell_3)&=\tfrac9{10}\,2+\tfrac1{10}\,0=\tfrac95,\\
U(\ell_4)&=\tfrac35\,0+\tfrac25\,2=\tfrac45.
\end{aligned}
$$
Since $12/5>9/5>3/2>4/5$, this <utility function> gives exactly the required <strict preferences>.
The distinction is that the <expected utility hypothesis> does not by itself require the <utility function> to increase with monetary amount. This <nonmonotone monetary utility> ranks the outcome paying 2 below the outcome paying 1. If increasing monetary <utility> were imposed as an additional assumption, the stated order would fail: $\ell_3\succ\ell_4$ requires $u(1)>u(2)$ because
$$
U(\ell_3)-U(\ell_4)=\tfrac12\bigl(u(1)-u(2)\bigr).
$$
That additional monotonicity assumption is not printed in this part.
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