Hi folks. My class (not a probability course) just covered a bunch of probability in one class period. Last prob course I had was in early high school so I'm a little fuzzy on it.

Here's an example we have to work on:

Everyone goes to a party with a gift and gives it to the secret santa, who passes them back to everyone randomly (in a random order). If n people go to the party, what is the expected number of party-goers that will get their own gifts back?

I need to use Indicator Random Variables for this. So here's what I have.

$\displaystyle \mbox {S is a sample space } \\ \mbox {E is an event of S}$

So each person either gets the gift or does not.

SO, first thing's first, I need to identify my IRV C_i

$\displaystyle C_i = \left\{\begin{array}{cc}0, & \mbox{ if they get a real gift } \\1, & \mbox{ if they get back their own gift } \end{array}\right.$

Next, since C is the random variable that represents # of people who got back their own gifts...

$\displaystyle C = \begin{array}{cc}n\\ \sum \\ n=1\end{array} {C_i}$

Then, getting a good gift back is:

$\displaystyle Pr[{C_i}=0] = \frac{1}{2}$

And getting your own gift back is:

$\displaystyle Pr[{C_i}=1] = \frac{1}{2}$

So our expected value of C_i is:

$\displaystyle E[{C_i}] = 0 * Pr[{C_i =0}] + 1 * Pr[{C_i =0}] = 0 * 1 * \frac{1}{2} = \frac{1}{2}$

And finally, the expected value of C is...

$\displaystyle E[C] = E[ (\begin{array}{cc}n\\ \sum \\ i=1\end{array}) {C_i} ]$

$\displaystyle E[C] = \begin{array}{cc}n\\ \sum \\ i=1\end{array} E[{C_i}]$

$\displaystyle E[C] = \begin{array}{cc}n\\ \sum \\ i=1\end{array} \frac{1}{2} = \frac{n}{2}$

(I hope you like my latex, I really tried to learn that for this post. )

So is this right? Any advice? Thanks!