This paper describes an extended ticket lottery model with an upper bound on the number of tickets and how to maximize the probability of success. The lottery works on the basis that there will always be one and possibly two winners each round even though there are more tickets in the pool than the number of tickets distributed to the participants.
It can be compared to a competition where participants guess a number and the winner is the one who made the best guess. This way the winner does not necessarily have to guess the exact number, but must be the one who guessed the closest number. If the range within where the winning number is chosen is known in advance and nobody can is allowed to pick the same number the probabilities are as described in this paper (probably).
The method of maximizing the probability of success was intended to be used for the type of ticket lotteries being run in the online game Jottonia.
The beauty (read: inherent uglyness) of these results is how Equation 3 takes care of calculating unused tickets covered by a picking the ticket in the middle of a gap. Honestly, there must be a simpler formula which does the same job.
NOTE: This post has been imported from my old it’s learning ePortfolio readables blog.