Expected Heads in Twenty Flips

Write the total number of heads as a sum of 20 indicator variables, one per flip.

Each indicator has expectation $1/2$; sum them using linearity of expectation.

1. 1. Set up indicators

   Let $I_k=1$ if flip $k$ is heads. Then the total number of heads is ${\sum }_{k=1}^{20}{I}_k$, and $E[I_k]=1/2$.

2. 2. Apply linearity

   Linearity of expectation holds regardless of independence.

   $$E\left[\sum _{k=1}^{20}{I}_k\right]=20\cdot \frac{1}{2}=10$$

Answer: 10