Q.7.11 - A First Course in Probability

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. 

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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

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

Step 3: 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

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

$=(n-1)·2p(1-p)$.

Step 4: Final Answer

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