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The fair premium in lottery games can be defined as the expected pay-off. For example, consider the game where you roll the die once and you get paid the face value in dollars. So, if you roll 1 you get $1, if you roll 2 you get$2, and so on. Then, the fair price to enter the game is the expected payoff which is:

$$E[X] = \sum_{i=1}^{6} x \times p(x) =$$

$$= \frac{1}{6}\times1 + \frac{1}{6}\times2+ \frac{1}{6}\times3+ \frac{1}{6}\times4+ \frac{1}{6}\times1+ \frac{1}{6}\times5+\frac{1}{6}\times6 = 3.5$$