Based on comments by cellocgw I decided to look at last week’s Secret Santa again. This time, the moment a person, whoever that is, draws his/her own name, the drawing starts again at the first person.

Introduction

A group of n persons draws sequentially names for Secret Santa. Each person may not draw his/her own name. If a person draws his/her own name then all names are returned and the drawing starts again. Questions are such as: How often do you draw names?

Simulation 

To understand the problem a simulation is build. The simulation draws the names, with restarts if needed. These are placed into outcome. On top of that it counts where a person self-draws.
selsan1 <- function(persons) {
  startoutcome <- rep(0,persons)
  countstop <- rep(0,persons)
  outcome <- startoutcome
  done <- FALSE
  possible <- 1:persons
  who <- 1           
  while(!done) {
    remaining <- possible[!(possible %in% outcome)]
    if (length(remaining)==1) {
      if (who==remaining) {
        countstop[who] <- countstop[who]+1
        outcome <- startoutcome
        who <- 1
      } else {
        outcome[who] <- remaining
        who <- who+1
        done <- TRUE
      }
    }     else {
      select <- sample(possible[!(possible %in% outcome)],1)
      if (who==select) {
        countstop[who] <- countstop[who]+1
        outcome <- startoutcome
        who <- 1
      } else {
        outcome[who] <- select
        who <- who+1
      }
    }
  }     
  return(list(outcome=outcome,countstop=countstop))
}
In an unlucky draw for five persons there were four restarts, at person 1, 2, 3 and 5.

Calculations

Chance of succes

Where fall the redraws