4 sprint segments, 2 laps, totaling 8 power ups. What do I get? 6 freaking ghosts.
There are 3 power ups, with supposedly 1/3 chance for each. The probability of getting 6 ghosts and 2 of something else is 1/31/31/31/31/31/32/3*2/3=22.4%. This is not a trivial chance to be at a disadvantage when someone is popping their van or aero power up in a sprint race.
Two culprits here:
Ghost is under tuned. It is simply not long enough to make a difference in a large group.
Each event of power up allocation is independent. So the probability of getting 8 ghosts out of 8 gates is the same as getting 8 aero power up out of 8 gates.
If for some reason we don’t want to buff ghost power up, at least make the power up events in races dependent on each other. For instance, if there are 3 power ups available in a race and someone has gotten the same power up 3 out of 9 times already, remove that power up from the bucket they can draw from for the remainder of the race. I know this is against the assumptions of probability theory, but screw the theory man, I don’t want to see ghost or burrito 6 times in a row in a race.
I feel you about getting bad power ups. But those odds you calculated mean that more than 1 in 5 racers are getting the same deal, right? That’s not that bad given a system that is designed to include that random factor.
What might be interesting is if it wasn’t randomized at all. Give everyone the same power ups, in the same order, and let people manage them as they want.
I don’t know what else went wrong with your calculation, but you haven’t multiplied by the appropriate binomial coefficient (8C2 = 28) to represent the different orders in which the ghosts and non-ghosts could occur.
Your actual probability of getting 6 freaking ghosts and 2 others is 8C2 x (1/3)^6 x (2/3)^2 = 1.71%
If using the binomial distribution with successes being ghosts then 8C6, although it equals 8C2 = 28, but in general the multinomial distribution would probably be better for multiple powerups, or perhaps the hypergeometric if they were to sample without replacement as per your original suggestion.
I don’t think combination applies here. I am not drawing 6 power ups all at once. They are all independent events. We are rolling the dice at every gate basically. If all numbers on the dice were equally useful for a sprint race, that’s no big deal. But currently, they are not.
Is the calculation above actually relevant? My understanding of the power up distribution is that it would be across all participants, not just a specific rider. So, without knowing what everyone else received, can it really be determined if the distribution was not 1/3 to each PU?
This is really the problem, I think. Race Directors enabling powerups that aren’t going to be useful for the particular race. I suspect you wouldn’t have been much happier if they’d had Feathers enabled and you’d gotten 3 of those instead of 3 of your Ghosts. If all the enabled powerups were helpful in that race, different story right?
I think so. But even with inequality of power ups, there is a way to prevent people from having really subpar situations relative to other racers. If I’ve already gotten 3 of the same power up in a race that I can get a total of 9 power ups, don’t make that power up available to me anymore for the rest of the race. This is not hard to code honestly. Distribution of power ups approaches their inherent probabilities only in large samples, across many many races I race. But from a personal experience standpoint, nobody cares for that. We want to see that we get a fair distribution of power ups per race.
Everybody might have gotten all draft powerups in the same race. Someone else’s chances of getting a power up is independent of yours, if it is truly random.
The issue stems from the fact that there will never be enough gates (approximately infinite) for the percentage of power ups someone receives to be roughly equal. In a race, we always have a very small sample, thus the number of heads and tails (figuratively speaking) will always be lopsided.