Let say we play a game. The game which spans the course of 5 trials. The game is as follows. You either receive 100 points or 40 points as a final payout. The only time a payout of a 100 points occurs (this is the real prize payout) is on the trial 5. Any unexpected payout until trial 5 is unfortunately just 40 points. If on any trial you do not get 40 points, you earn 5 points as a reward. Remember our goal is receive that final 100 points on trial 5. Question, how much should we pay to play such a game with unknown possible payouts.
So here are all the possible scenarios for this game
40 Failure Trial 1
5 40 Failure Trial 2
5 5 40 Failure Trial 3
5 5 5 40 Failure Trial 4
5 5 5 5 40 Failure Trial 5
5 5 5 5 105 No Failures at any Trial, Final payout of 100 occurs
Lets call the value of the gamble is V. So Given all these payoffs. The value of the gamble should be the E(V), the expected value of all the possible payoffs. But what probability density should I use to weight the possible scenarios? I could assume that on any given trial the probability that a 40 point pay out occurs is 2%. Hence the probability we get to that 40 point payout on our 1st trial is 98%, the probability we get to that 40 point payout on our 2nd trial (98%)^2, 3rd trial (98%)^3,.. etc. Another way to view it is based as to when the low probability 40 point payout actual occurs.
(.02)
(.98)(.02)
(.98) (.98)(.02)
(.98) (.98) (.98)(.02)
(.98) (.98) (.98) (.98)(.02)
(.98) (.98) (.98) (.98)(.98)
But obviously these probabilities do not produce a pdf because they do not sum to 1. Obviously the probabilities listed above do not include all the real of possibilities because we are dealing with conditional probabilities. What is the correct pdf to value these payouts?