[lahvak]

A mandatory nitpick: you cannot do conditional probability without multivariate distributions, but you are right, you do not need linear algebra for a basic introduction. Even continuous distribution can be introduced, you just start with discrete cases, introduce areas in a histogram, and then approximate the histogram by a continuous curve. You can even get to the Riemann integral that way if you want to. And nobody actually calculates normal probabilities by integrating the Gaussian function, it's kind of hard to do.