Did you get your fair share of the cake?

Probability

Are all proportional divisions envy-free? Are all envy-free divisions proportional?

All proportional divisions are envy-free, but not all envy-free divisions are proportional. All envy-free divisions are proportional, but not all proportional divisions are envy-free. All proportional divisions are envy-free and all envy-free divisions are proportional. None of these statements is true.