The basic problem we study in probability:

Given a data generating process, what are the properties of the outcomes?

The basic problem of statistical inference:

Given the outcomes, what can we say about the process that generated the data?

(ref: Wasserman(2004))

For example:

1. Given that we know (assume) that the IQ test is scored so that it has a normal distribution with the mean score of 100 and a standard deviation of 15, what percent of the population has a score higher than 130?
2. Given that we collect IQ score data on 1000 randomly chosen individuals, what is our best guess estimate of the mean and the variance of the normal distribution in the underlying population that we believe generated the data.