Hypergeometric distribution is a fancy formula people can use to determine, if we pick a number of things from a group without replacement, the probabilities of how many things we'll get. This can also be used to determine the probability of drawing a card from your deck in by turn X, so there's some fun for you.
We have 22 people, our population size.
4 of which are mafia, our successes,
We're picking 5 people at random, our sample size,
And we're checking to see the probability of having one mafia in the five we pick, which is the number of successes in our sample.
If the mafia kills one first, we have 21 people, not 22. This EVER so slightly increases your chances. The difference is in the spoiler below.
If we randomise now, and then the mafia kill one not in the group, we've done 5 picks from a sample size of 22, not 21, which is slightly inferior.
If we randomise now, and then the mafia kill one in the group, we've done 4 picks from a sample size of 22, then a new sample from 21... Or is it 18? Not sure how it works. e.e
Here are the exact probabilities for the gamblers that care: