Probability

Every probability is a number in the interval from 0 to 1, and the probabilities of all the outcomes in the sample space add to exactly 1. $P(A)$, read "the probability of A", is the long-run relative frequency of the event $A$: the share of trials on which $A$ happens, once the number of trials is large. When every outcome is equally likely you can reach that number by counting, dividing the outcomes in $A$ by the total, but counting is a special case rather than the definition.

Draw a single digit at random from 0 through 9, each with probability 0.1. Let $A$ be "the digit is 0, 1, or 2". Then $P(A) = 3/10 = 0.30$.

"A probability of 0.30 means three out of every ten." Read ten digits from a random digit table and the chance that exactly three of them fall in $A$ is 0.2668, so that reading is wrong about 73 percent of the time, and roughly 2.8 percent of ten-digit runs contain none from $A$ at all. A probability is a rate the results approach over a long run, not a quota each block of ten has to fill. What the long run does and does not promise is the content of the law of large numbers.

Two boundaries are worth naming. Counting outcomes gives the right answer only when the outcomes really are equally likely, which rules out a loaded die or a thumbtack. And probability 0 does not always mean impossible: for a continuous random variable every single exact value has probability 0, yet the variable still lands somewhere.

Probability is introduced in topic 2.4, Introduction to Probability, after simulation has already given you an empirical version of the same quantity.