# Probability of dataset given normal distribution parameters

I am reading through Christopher Bishop's Pattern Recognition and Machine Learning book and came across this part that I did not quite understand. As I understand it, this is claiming that the probability of a set of data is equal to the product of the p.d.f. of the normal distribution of the datapoints. However, the p.d.f. can be greater than one, especially if $\sigma$ is small. So if it happens that I sample a large amount of my data near the mean, isn't it possible to return a value greater than 1 for the probability of the dataset? I feel I have a misunderstanding somewhere since this shouldn't be possible.

However density is proportional to a probability. e.g. in the univariate case the probability of being in the interval $(x,x+\delta )$ would (as $\delta$ becomes very small) approach $p(x)\delta$ (this may be put more formally but my aim here is not to be formal).