Equally likely outcomes

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.