Expectation as a fair price
- Expectation ~ weighted average (over probability) of possible outcomes/values
- Probability (say of a Bernoulli random variable, a coin toss, X=1 if head and X=0 if tail) as a probability
$$P(X=1) = E(X) = 0 P(X=0)+ 1 P(X=1).$$ - Long-run frequency interpretation of a probability
- Random vending machine: 是否可接受這樣的販賣機取決於你是否是一個經常使用者。一次，甚至少數次，的使用不見得可以感覺到機率的作用。
- Gambler’s ruin theorem
- St. Petersburg’s paradox