B Does the binomial distribution play a role determining p from data?

[benorin]

TL;DR Summary

Trying to deduce the dodge rate from given data using binomial distribution somehow, not sure how to proceed

In a game heroes have a maximum dodge rate, from experimental data we have 13 dodges out of 24 attacks (so 11 hits). A fellow on my discord server had immediately solved for the dodge rate as being 13/24. I started to explain it is not so simple as dividing (24-11)/24=13/24 is not the dodge rate, this experiment follows a binomial distribution with number of trials n=24, number of successes x=13, and probability of success p=dodge cap (in decimal form): but here’s where I got lost plugging in different dodge rates into a binomial calculator and observing various probabilities for the experiment to have run the way it did for different values (we have it on a good word that the dodge cap is 50 or 60%) and realized this approach did not provide me a means of determining the actual dodge cap. Have I gone and made this overly complicated? Was 13/24 actually a good estimate? Thanks!

 

[Dale]

It sounds like you want to do a Bayesian estimation of the range of possible probabilities based on the observed data. The first thing is to determine your prior probability.

 

[WWGD]

Under some conditions the binomial can be approximated well by a normal with mean np and variance np(1-p) (But you use formulas for sampled data). Maybe you can use this to construct a confidence interval, or ate you looking for a point estimate?

 

[Dale]

(we have it on a good word that the dodge cap is 50 or 60%)

So, if you want to do a Bayesian approach the first thing would be to start with this. How solid do you think this "good word" is? Do you think it is 50% likely to be true, or 80%, or what?

Once you have established how reliable you think this prior knowledge is then you can construct a Beta Distribution that matches that. For example, if you think that the prior is 50% likely to be correct then you could use $\beta(25,20)$ which has only 25% probability of being less than 0.5 and only a 25% probability of being greater than 0.6.

Then, to account for your data you would simply add the 13 successes and 11 failures to your Beta Distribution to get $\beta(38,31)$. That would be the posterior distribution of the dodge cap accounting for both your prior information and also your data.

Here is the resulting plots of your prior (blue) and posterior (orange) probability distribution curves.

 

[Stephen Tashi]

Was 13/24 actually a good estimate?

If we take "good estimate" to mean "what many statisticians would have done" then yes, it was a "good" estimate.

However, if you are going to keep encoutering problems in probability and statistics, you should learn the basic scenario for the theory of statistical estimation. This would include the various technical meanings of "good estimate" and the various ways of making and analyzing estimators.

The mean of the binomial distribution $B(N,p)$ is $Np$. Setting the observed number of successes $k$ in a particular sample equal to $N\hat{p}$ and solving for $\hat{p}$ defines one possible estimator for $p$. Whether this estimator is "good" or not is something that can be investigated if you define which properties of estimators interest you.As to other possible estimators, the Bayesian approach mentioned by @Dale can be used to incorporate information such as "p is 50% or 60%". It's interesting that the original paper by Rev. Bayes considered the general problem that you are considering. https://en.wikipedia.org/wiki/An_Essay_towards_solving_a_Problem_in_the_Doctrine_of_Chances