MathProblem 44 Coin toss problem #1

You are given an initial prize of one dollar. A coin will be tossed indefinitely until a tails occurs. Every time a heads is tossed your prize will be doubled. When a tails is tossed you walk away with whatever your prize has grown to. For example if you toss three heads and then tails you would get \(1*2*2*2=8\) dollars. What is the expected return of this game?

Solution

每次扔到 \(head\)， 价值会 \(double\); 如果是 \(tail\)，则结束。现在要求期望的报酬。

\[\begin{align} E[X] &= \sum_{x=0}^{\infin}\frac{1}{2}\cdot (\frac{1}{2})^x\cdot 2^x\\ &= \infin \end{align} \]

其中 \(x\) 表示 \(x\) 次 \(head\)，最后一次 \(tail\) 的概率