EV Indicator RV: Married Couples
Nine married couples are standing in a line. They mix up and form nine pairs at random. What is the expected number of couples that end up paired together?
Answer
Let \(X\) be the number of couples paired together. For each couple \(i\) (\(i=1,\ldots,9\)), define the indicator variable
\[
X_i = \begin{cases}
1, & \text{if couple \(i\) is paired together},\\[1mm]
0, & \text{otherwise}.
\end{cases}
\]
Then
\[
E[X] = \sum_{i=1}^{9} E[X_i].
\]
Since \(P(X_i=1)=\frac{1}{17}\), we have
\[
E[X] = 9\cdot\frac{1}{17} = \frac{9}{17}.
\]