后续正面次数

为了找到接下来 n 次抛掷中正面朝上的预期数量，我们需要知道选择了哪枚硬币。使用两次抛掷都是正面的条件，我们可以计算每枚硬币的条件概率。出现两个正面的概率是每枚硬币出现两个正面的概率的并集。 $$P(2H) = \frac13(\frac13)^2 + \frac13(\frac23)^2 + \frac13(1)^2 = \frac{14}{27}$$ $\frac13$ 硬币被选中的概率为： $$\frac{\frac13(\frac13)^2} {\frac{14}{27}} = \frac{1}{14}$$ 同样，$\frac23$ 硬币被选中的概率为： $$\frac{\frac13(\frac23)^2} {\frac{14}{27}} = \frac4{14}$$ 留下总是正面硬币被选为 $\frac9{14}$ 的概率。现在我们知道了每枚硬币被选中的概率，我们就可以求出接下来 12 次抛掷中正面朝上的预期数量。如果我们有一枚硬币正面朝上的概率为 $p$，则我们可以预期 $12p$ 为正面。因此预期的头数为： $$\frac1{14}(12\cdot\frac13) + \frac4{14}(12\cdot\frac23) + \frac9{14}(12\cdot1) = \boxed{\frac{72}7}$$

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Original Explanation

To find the expected number of heads in the next n flips, we need to have an idea of which coin has been selected. Using the condition that both flips were heads, we can compute the conditional probability of each coin. The probability of two heads is the union of the probabilities of getting two heads for each coin. $$P(2H) = \frac13(\frac13)^2 + \frac13(\frac23)^2 + \frac13(1)^2 = \frac{14}{27}$$

The probability the $\frac13$ coin was selected is: $$\frac{\frac13(\frac13)^2} {\frac{14}{27}} = \frac{1}{14}$$ Similarly the probability the $\frac23$ coin was picked is: $$\frac{\frac13(\frac23)^2} {\frac{14}{27}} = \frac4{14}$$ Leaving the probability the always heads coin is picked to be $\frac9{14}$. Now that we know the probability of each coin being selected, we can find the expected number of heads in the next twelve flips. If we have a coin with probability $p$ to land on heads, we can expect $12p$ heads. Thus the expected number of heads is: $$\frac1{14}(12\cdot\frac13) + \frac4{14}(12\cdot\frac23) + \frac9{14}(12\cdot1) = \boxed{\frac{72}7}$$