MarioKart Wii WFC Probability?

[moonman239]

So I was playing MarioKart Wii on the Internet yesterday. In case you're not familiar with the game, it's a game where you select a Nintendo character (such as Mario, Luigi, Peach, or Bowser) and a car and can race against other characters (or other people). Each player involved in an Internet game can either vote for a racetrack or choose a random track. After all players make their choice, the game proceeds to randomly select a course that has been voted for (or a random course). Which poses an interesting question: if all player's choices are unknown, what is the probability that a particular course will be voted for by the player, and will be chosen by the game?

[brian44]

Assume there are k tracks, giving a total of k+1 choices (player may choose random). Assume the computer selects one of the tracks, or random, with probability according to the number of votes/total for each, call the number of votes for track i v_i, i =1,..,k, number of votes for random v_r, and total number of votes v_t. Further assume that if the computer selects random, then each track has equal probability of being selected.

Then we have
P(selected track = track i) = v_i/v_t + (1/k)v_r/v_t

[cuak2000]

"if all player's choices are unknown" ->
Does that mean we should assume a uniform distribution?