Adjacent Equal Coin Flips

There are $19$ adjacent pairs among $20$ flips.

Each pair matches independently with probability $1/2$; use linearity of expectation.

1. 1. Set up indicators

   For $i=1,\dots ,19$, let $I_i$ indicate that flip $i$ and flip $i+1$ are equal. Each has $P(I_i=1)=1/2$.

2. 2. Sum by linearity

   Summing over the $19$ adjacent pairs.

   $$E[X]=\sum _{i=1}^{19}E[I_i]=19\cdot \frac{1}{2}=\frac{19}{2}$$

Answer: 19/2