By the POSTULATES OF PROBABILITY. Meaning of "it is always assumed that one of the possibilities in S must occur"

Solution 1:

From reading the comments and your post itself, I think you have been mixing up the terminology. An illustration may clarify:

Randomly choosing one ball from each of four bags, each containing one black and one white ball, each equally likely to be chosen, is a $4$-trial probability experiment with $16$ outcomes.

So, this experiment's sample space, which comprises the experiment outcomes, is $$\{BBBB,BBBW,BBWB,BBWW,\\BWBB,BWBW,BWWB,BWWW,\\WBBB,WBBW,WBWB,WBWW,\\WWBB,WWBW,WWWB,WWWW\}.$$

An event is simply some subset of the sample space.

So, this experiment has $2^{16}=65536$ possible events, including the empty set (i.e., an impossible event, e.g., ‘choosing a yellow ball’), the sample space itself (i.e., a certain event, e.g., ‘choosing four balls’), and any combination of the $16$ outcomes, e.g., $\{BWWW,WBWW,WWBW,WWWB\}=$‘choosing exactly one black ball’.

In particular, an elementary event contains just one experiment outcome, so this experiment has $16$ elementary events, e.g., $\{WWWW\}.$

Read more here.

OP: Could we say that either at least one or exactly one of THE $16$ OUTCOMES in the sample space must occur"?

Good question. In any given experiment, exactly one outcome occurs; in our experiment, the outcomes $BBBB$ and $BBBW$ cannot simultaneously occur.

However, in any given experiment, the outcome $BBBB$ can eventuate via $2^{15}=32768$ (out of the $65536$) different events—e.g., $\{BBBB\},\{BBBB,BBBW,WWWB\}$—each of which has a probability.

It's worth pointing out that

• “the probability of outcome $BBBB$” is actually a shortening of “the probability of the event of outcome $BBBB$”,
• writing $P(BBBB)$ is just abusing notation to mean $P(\{BBBB\}),$ and
• $P(BBBB,BBBW,WWWB)$ (continuing to abuse notation) means $P(BBBB \text{ or } BBBW \text{ or } WWWB),$ not $P(BBBB \text{ and } BBBW \text{ and } WWWB).$

OP: The terminology I have been mixing includes the word one in "one of the possibilities in S must occur", and according to your 65536 possible events. I might understand the meaning of one. Does it mean that one of 65536 possible events or 65536 possible subsets or 65536 possible outcomes must occur. Do I get it right?

1. Both lulu and yourself seem to be intermittently conflating 'event' (a set) and 'outcome' (an element of a set).

   Every subset is a set, and every set is a subset (of at least itself).

   You are also conflating ‘subset’ and ‘outcome’; they are not synonyms!

   Ensure that you can clearly distinguish among the four boldfaced words above; in the process, your confusion will clear up.

2. Consider $BBBW$ and $\{BBBW\}:$ one is an outcome (an element in the sample space), the other is an event (a subset of the sample space).

3. The event of choosing a yellow ball is the empty set $\emptyset,$ which contains none of the 16 outcomes; in other words, it is an impossible event and has probability $0.$ After all, to quote your book, "one of the possibilities in S must occur".

4. To return to your initial query: "one of the possibilities in S must occur" means "one of THE $16$ OUTCOMES in the sample space must occur".

   (To be clear: “possibilites” is just an informal word and could, in an alternately-constructed sentence within the same context, also refer to the $65536$ events.)