I was watching this video, in which a "professional astrologer" is tasked with matching up people with their star signs.
(Spoiler alert!) At around the 5:30 mark, 4 of the 12 participants reveal that their star sign was correctly identified.
This leads to the question: What is the probability of him getting 4 or more star signs correct, and can this be nicely generalised to getting $x$ or more correct answers out of a possible $y$ people?
Note that each of the twelve participants have a different star sign, and so do each of the 12 cards.
There is a similar question here - coincidentally inspired by the same series - however, this focuses on the expected value rather than the general probability.
Thanks, Sam
Not an explicit answer, but some references.
What you're looking for is $X = $the number of fixed points in a random permutation. (I.e. if the astrologer guesses a random permutation, how many would he/she guess correctly?)
The distribution of $X$ can be obtained via the Inclusion-Exclusion Principle. Here is one reference that includes the formula for $P(X = k)$:
http://demonstrations.wolfram.com/TheNumberOfFixedPointsInARandomPermutation/
And https://en.wikipedia.org/wiki/Random_permutation_statistics contains a ton more details on related problems, but AFAICT not the explicit formula.