Let's say we have a list of all the chess players in the world, and we want to predict the likelihood of success if any player goes up against any other player. (Hypothetical example)
I'm assuming that it is possible for any player to beat any other player, regardless of skill. However, playing versus a player with much lower skill a loss is very unlikely, while playing versus a player with much higher skill makes it very likely to lose. Therefore, the mean probability of a loss should follow the S-shaped curve of a cumulative binomial probability distribution (if x = difference in skill level and y = probability of loss against a player of this skill), where each player has their own probability distribution like this and the center with 50/50 chance is playing against a player of exactly the same skill.
The thing is that the cumulative binomial probability distribution only describes the mean probability of a loss. I want to create a model where there is an actual distribution for any point along these means but with a gaussian distribution (or something such) so that the highest probability is along that mean but there is always a chance to win.
Example 1: Player of skill X goes up against player of skill X-10 and gets 1 loss and 9 wins, goes up against a player of skill X and gets 5 losses and 5 wins, goes up against a player of skill X+10 and gets 9 losses and 1 win. (most likely scenario)
Example 2: Player of skill X goes up against player of skill X-9 and gets 5 losses and 5 wins, goes up against a player of skill X and gets 2 losses and 8 wins, goes up against a player of skill X+9 and gets 3 losses and 7 wins. (very unlikely scenario, but possible)
The thing here is that the general format of the probability curve will be the same for all players of any skill, and the mean will follow the S-shape versus players of lower/same/higher skill... But a very skilled player will have less variability around that mean and the prediction will be more certain, whereas a lower skilled player will be more erratic and depend on "having a good day" or similar things. So I need a variable in the model that is connected with skill and controls the width of the probability distributions around the mean.
FYI if you are interested to discuss "the best way to predict chess winners" then this is not a thread for you, I am not actually interested in chess. I am a programmer with limited math skills who wants to find a way to predict the outcome of a skill matchup. The chess part was just a hypothetical example, it could literally be anything in my final model, if I arrive at a result with your help.
EDIT:
What I mean to say is that the means should follow a logistic function according to:
1 / (1 + exp(-skill * (x - 0.5)))
That way, there is always a 50/50 chance of loss when playing against someone with equal skill. A person with lower skill will have a higher chance of loss against someone with lower skill than themselves, and a lower chance of loss against someone with higher skill than themselves (because that person is also unskilled), compared to someone with higher skill. Exactly what the limits of the "skill" variable should be, I'm not quite certain of. But I can play around with that according to what value I choose to represent skill. Actually, maybe the skill value should be inversed, if the gaussian width is decreased. I'm not sure.
What remains to do is to somehow fit this mean to a gaussian probability with 0-1 limits.