Equally Likely Outcomes vs Probability

Outcomes are equally likely when the sample space holds $k$ outcomes and each one carries probability $1/k$. When that holds, $P(A)$ is the number of outcomes in $A$ divided by $k$ for every event $A$, so counting is enough to get a probability. When it does not hold, the counting formula has no warrant behind it. Equal likelihood is an assumption about the process, normally justified by physical symmetry. Nothing about writing down a list of outcomes makes it true.

A fair six-sided die has six symmetric faces, so $P(\text{roll above 4}) = 2/6 \approx 0.3333$. Load it so a 6 turns up with probability 0.30 and the other five faces split the remaining 0.70 evenly at 0.14 each, which still totals 1. Now $P(\text{roll above 4}) = 0.14 + 0.30 = 0.44$. Counting still returns 0.3333 and is simply wrong, because the counting formula was never a statement about this die.

"The sum of two dice runs from 2 to 12, so there are 11 possibilities and $P(\text{sum} = 7) = 1/11 \approx 0.0909$." Those 11 sums are a legal sample space, since exactly one of them happens on every roll, but they do not carry equal probability. The equally likely outcomes are the 36 ordered pairs of faces, and 6 of them total 7, so the answer is $6/36 = 1/6 \approx 0.1667$, nearly double. Equal likelihood belongs to the particular list you chose rather than to the process, and lumping outcomes into categories is what destroys it.

Two boundaries. Real processes rarely arrive with symmetry you can verify, so a thumbtack, a bent coin, or the first person through a mall door has no claim on equal probabilities, and the honest move is to estimate them from data or by simulation. And a continuous sample space has no outcomes to count at all, which is where a uniform distribution takes over, spreading probability evenly across an interval as area.

Counting equally likely outcomes is where topic 2.4 starts.

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.