Lotteries and expected utility Suppose we have the following four lotteries:

$L_{1}=[(1,\$1)]$

$L_{2}=[(0.01,\$0),(0.89,\$1),(0.1,\$5)]$

$L_{3}=[(0.9,\$0),(0.1,\$5)]$

$L_{4}=[(0.89,\$0),(0.11,\$1)]$

If our agent says that he prefers $L_{1}$ to $L_{2}$ and prefers $L_{3}$ to $L_{4}$, then he is not following the expected utility assumption of independence (also known as substitutability).

Anyone know how to explain this?
Your example is the classic Allais paradox. 

I think the best way to see how the preference pattern $L_1\succ L_2$ and $L_3\succ L_4$ violates independence is to visualize it geometrically. Consider the following figure: 



Observe that 
\begin{align}
L_2&amp;=L_1+(0.01,-0.11,0.1)\\
L_3&amp;=L_4+(0.01,-0.11,0.1).
\end{align}
Therefore, the indifference curves passing through $L_1$ and $L_4$ should be parallel. Since $L_1\succ L_2$, the latter lottery should lie below the IC through $L_1$. By implication of the parallelism of the ICs, $L_3$ should also lie below the IC through $L_4$. But we're given that $L_3\succ L_4$. This means that $L_4$ lies below the IC through $L_3$ (the blue dashed line). Since parallelism of ICs is a necessary condition of independence, the agent's preference is inconsistent with independence, because it can only be justified by a set of non-parallel ICs.

Alternatively, you can think of the inconsistency between the agent's preference and expected utility theory in the following way.
Suppose expected utility theory holds. Then $L_1\succ L_2$ is equivalent to the inequality
\begin{equation}
u(\$1)&gt;0.01u(\$0)+0.89u(\$1)+0.1u(\$5).
\end{equation}
Adding $0.89u(\$0)-0.89u(\$1)$ on both sides, we get 
\begin{equation}
0.89u(\$0)+0.11u(\$1)&gt;0.9u(\$0)+0.1u(\$5). 
\end{equation}
But this second inequality is simply saying $L_4\succ L_3$, which is contradictory to the agent's preference.