There are many Secret Santa problems. The one we're interested in here is this:

 Suppose 10 people decide to participate in a "Secret Santa". Their names are put into a hat, and one by one each takes a name out. If it's their own, they put it to one side, choose another, then return their own to the hat. They then retain the name they've chosen, which is clearly not their own. So all but the last are guaranteed to end up with a name other than their own. What is the probability that the last person ends up with their own name?
 Here's one analysis ... The process guarantees that the first 9 won't have their own names. How many arrangements are there? If the last person does have their own name then the first 9 have a derangement. If the last person does not have their own name then all 10 form a derangement. Total number is $d_{n-1}+d_n.$ Then the probability that the last person has their own name is $\frac{d_{n-1}}{d_{n-1}+d_n}.$ When n is very large, this approaches $\frac{1}{n+1}.$ The exact numbers are: For 5 people: 9/(9+44) = 0.1698... For 7 people: 265/(265+1854) = 0.125059... For 10 people: 133496/(133496+1334961) = 0.090909029... This turns out to be wrong! The different derangements are not, in fact equally likely.

So I've been doing some calculations, and here are the results:

 Number of participants Probability of failure Evaluated 1 1 1.000000 2 0 0.000000 3 1/4 0.250000 4 5/36 0.138889 5 19/144 0.131944 6 203/1800 0.112778 7 4343/43200 0.100532 8 63853/705600 0.090495 9 58129/705600 0.082382 10 160127/2116800 0.075646 11 8885501/127008000 0.069960 12 1500518539/23051952000 0.065093

These results match exactly those of the simulations I've been running, so I know they're right. They are strange numbers, though, so where do they come from?

Being so strange you might imagine that the calculations aren't simple or straight-forward, and they're not. The calculation for N=1, 2 or 3 aren't very enlightening. Here's the full calculation for N=4.

We're choosing from A, B, C and D. A can't go first, so we have three choices, and they're equally likely. Here's a table for the next choice:

 Chosen Next choices B A, C, D C A, D D Don't care

In the last case we don't care because we're trying to compute the probability of the process failing. With D selected it can't fail.

But now we have 5 ways to continue: BA, BC, BD, CA, CD, and their probabilities are not equal:

 Chosen Prob Next BA 1/3 * 1/3 D BC 1/3 * 1/3 A, D BD 1/3 * 1/3 A CA 1/3 * 1/2 B, D CD 1/3 * 1/2 A, B

We only care if we get D last, so the only options we get are BCAD and CABD. The probabilities of these are $\frac{1}{3}*\frac{1}{3}*\frac{1}{2}$ and $\frac{1}{3}*\frac{1}{2}*\frac{1}{2}.$

Now we can see that the probability of failing is $\frac{1}{18}*\frac{1}{12}$ which is $\frac{5}{36}.$