MathProblem 61 Coin toss problem #3

Suppose you have a coin in which the probability of flipping a heads is \(p\), where \(p\ge0.5\) . What is the expected number of flips it will take for the number of heads to equal the number of tails, assuming the first flip is a tails?

Solution

将其转换为 \(random\ walk\), 其中向左走 \(+1\), 向右走 \(-1\), 那么期望的移动长度：

\[E[X] = p\cdot 1+(1-p)\cdot (-1)=2p-1 \]

那么问题转换为回到原点需要的期望步数是多少。由于一开始为 \(tail\), 所以起始为 \(-1\) 的位置，即：

\[E[N]\cdot (2p-1)=1\Rightarrow E[N]=\frac{1}{2p-1} \]