Suppose I have 1 heads and 4 tails from 5 coin tosses. To find out the probability of 1 heads and 4 tails in my coin toss experiments, I decided to use Binomial Probability Mass Function for the calculation of the probability on the current observations.
I used Maximum a Posterior with the Beta prior $ (\alpha=5, \beta=5) $, instead of Maximum Likelihood Estimation, to estimate the parameter value of $ \theta $ and I got $ 0.3846 $ from Maximum a Posterior. Now, I have the parameter value of $\theta$ and I want to find out the probability that I would observe 1 heads and 4 tails, which one of the following equations should I plug in the estimate parameter value, $ 0.3846 $:
- $ {n}\choose{k} $ $ \Pi^n_{i=1} \theta^{x_i} (1-\theta)^{(1-x_i)} \theta^{(\alpha-1)} (1-\theta)^{(\beta-1)}$
The above equation is considering the prior probability for the calculation of the probability for my experiment. Or should I just plug the value of $ \theta $ into the Binomial Probability Mass Function?
- $ {n}\choose{k} $ $ \Pi^n_{i=1} \theta^{x_i} (1-\theta)^{(1-x_i)}$
I know this sounds very naive but I just want to make sure I am thinking correctly.