Sorry, I'm a huge statistics noob. I'm looking to find the probability that a certain event occurs in a binomial experiment. I take a few thousand samples per test, and then calculate the probabilities. The problem is that the experimental probabilities keep varying by a small amount. So I've started plotting the experimental probabilities as a function of sample size. When I look at the plots for these "cumulative averages" the trend I observe is that the experimental probabilities vary wildly at first and then taper off and oscillate around the true probability. However, I can't get enough samples per test to observe a straight line-like convergence.
My question: can I average out these cumulative averages I've plotted for a more precise estimate of my true event probability? And if I can, would it be better to use a weighted average - like give more weight to probabilities using a higher number of samples and lower weight to probabilities using a lower number of samples? Or maybe even take the average of only the experimental probabilities that begin to converge very finely. Thanks