Confusion related to scaling factors in HMM

I was reading about HMM in C.M. Bishop's book Pattern Recognition and Machine Learning. I was going through the forward and backward algorithm using $\alpha$ & $\beta$

For forward messaging passing

$\alpha(z_n) = p(x_n|z_n)*\sum_{z_n-1}\alpha(z_{n-1})*P(z_n|z_{n-1})$

Since the probabilities are <=1. As we move from $z_1$ to $z_N$ the $\alpha(z_n)$ tends to get smaller and smaller and we might have floating point issues. For that the book mentioned something called scaling factors where u normalize the $\alpha$

So we have

$\hat{\alpha(z_n)} = p(z_n|x_1...x_n) = \alpha(z_n)/p(x_1...x_n)$

which we expect to be well behaved numerically because it is a probability distribution over K variables for any value of n. I didn't get how it is well behaved and how it solves the floating point issue and what does it mean by probability distribution over K variables. I just didn't get this part