I particularly like the Mermin paradox (also known as GHZ paradox), because you can understand it without knowing anything about probability theory.
Basically it goes as follows:
You have a specific entangled state of three particles, called GHZ state. On each of the particles you can do one of two measurements, $X$ and $Y$, and in both cases, you may get either $+1$ or $-1$ as result. Of course, you can choose independently on each single particle of the state whether you measure $X$ or $Y$ on it.
If you look at the measurement results of any single particle of the state, you find that you have randomly either $+1$ or $-1$. However if you look at a full set of measurements, you'll notice a pattern:
Whenever you measure $X$ on exactly one of the entangled particles, and $Y$ on the other two, you'll notice that the product of all three measurement results is always $-1$. This is the case no matter for which of the three particles you measure $X$.
Now this would not yet be a problem: It is compatible with the assumption that each measurement value is predetermined. Be $x_i$ the measurement result of measuring $X$ on particle $i$, and $y_i$ the measurement result of measuring $Y$ on particle $i$. Then the above fact means that we have the three equations $x_1 y_2 y_3 = -1$, $y_1 x_2 y_3 = -1$ and $y_1 y_2 x_3 = -1$.
Now we can just multiply those three terms together and using the fact that each $y_i$ is either $+1$ or $-1$, and thus $y_i^2 = 1$, we get:
$$x_1 x_2 x_3 = x_1 y_2 y_3 \cdot y_1 x_2 y_3 \cdot y_1 y_2 x_3
= (-1) \cdot (-1) \cdot (-1) = -1$$
Therefore we would expect that if we measure $X$ on all three particles, we also find that the product of the three values is $-1$.
However, quantum mechanics tells us, and experiment confirms (within measurement error), that if we measure $X$ on all three particles, the product of the three measurement results always is $+1$.