And I really want to hit that point home because sometimes in probability books they'll just give you a formula-- oh, the expected value of a probability distribution is each of the outcomes times their frequency.

我真的很想说清楚这一点，因为有时在概率书中他们只会给你一个公式——哦，概率分布的期望值是每个结果乘以它们的频率。