In Bayesian statistics does posterior probability become prior probability when new observation is made? First you have prior probability say 50% . When observation is made we use that to update our prior probability = posterior probability 66%. If another observation is made our prior probability 66% which we would update with new observation right? and we continue this process?
I just want to clarify if my understanding it correct.
 A: Just one thing, prior is a probability distribution, not a value like 50% (prior may be also improper.) 
As an example with the Binomial and Beta prior: 
Starting with a prior distribution $Be(1,1)$ (this is the uniform distribution).
We see 5 outcomes, where 3 are positive and 2 negative.
The posterior is now $Be(1+3,1+2)$ = $Be(4,3)$. This is the prior for the next time.
We see another 13 outcomes, 8 positive, 5 negative.
The posterior is now $Be(4+8,3+5)$ = $Be(12,8)$.
We see 125 outcomes, 80 positive, 45 negative.
The posterior is now $Be(12+80,8+45)$=$Be(92,53)$.
Here is each curve.

A: Yes --- The terms "prior" and "posterior" are always relative to the evidence you are considering, so if you have already incorporated previous evidence and are now considering new evidence, then the "posterior" from your previous analysis becomes the "prior" for your present analysis.
One of the properties of Bayesian updating for a sequence of data is that it can be done equivalently either by incorporating the data "all at once" or by incorporating the data sequentially with each posterior forming the prior for the next recursive update (see e.g., these notes by Lauritzen 2009, esp pp. 1-4).
