Question

Consider $n$ independent flips of a coin having probability $p$ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $n=5$ and the outcome is$HHTHT$, then there are $3$ changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of $n-1$ Bernoulli random variables.

Short answer

The expected number of changeovers is$(n-1)2p(1-p)$.

Step-by-step solution

Given Information

Let $n$ independent flips of a coin have probability $p$of landing on heads.

Explanation

Define indicator random variables $I_j$that marks if there was a changeover between $jth$and $(j+1)$st flip, $j=1,...,n-1$. Observe that $I_j=1$if and only if we have Head in $jth$flip and Tail in $(j+1)$st flip or if we have Tail in $jth$flip and Head in $(j+1)$st flip.

So, we have that

$P\left(I_j=1\right)=2p(1-p)$.

Explanation

Define $x$as the total number of changeovers. We have that $X=\sum _{j=1}^{n-1}{I}_j$and using the linearity of the expectation, we have that

$E\left(X\right)=\sum _{j}E\left(I_j\right)=\sum _{j}P\left(I_j=1\right)$

$=(n-1)·2p(1-p)$.

Final Answer

The expected number of changeovers is$(n-1)2p(1-p)$.