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I read this from my textbook.

POSTULATE 2: P(S)=1.

It is always assumed that one of the possibilities in S must occur, and it is to this cetain event that we assign a probability of 1.

It seems to me the meaning of "it is always assumed that one of the possibilities in S must occur" is hard to understand.

Does it mean the certain event is the the sample space?

After all, any event is a subset of the sample space, and the subset could be the sample space.

Stats Cruncher
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    Not sure what you mean by the certain event. Yes, it is certain that whatever occurs is in the sample space, that's what the postulate means. – lulu Nov 20 '21 at 13:43
  • Thanks. But why is P(S)=1 explained in this way as descibed above? Is it correct to say that "*some possibilities must not occur so only that one possibility must occurs*"? – Stats Cruncher Nov 20 '21 at 14:26
  • Well, presumably, for any set of events $S$, you define $P(S)$ to be the probability that some event in $S$ occurs then, since we know that some event in the sample space must occur, of course the probability of the entire sample set is $1$. – lulu Nov 20 '21 at 14:29
  • Think of the sample space as a list of all the possible outcomes. That mental picture gets cloudy when you have continuous distributions, but I think the intuition still holds. – lulu Nov 20 '21 at 14:32
  • I don't understand your statement in bold. Deterministic events may be thought of as probabilistic events with only one possible outcome. In that case, there need be no possibilities which do not occur. Not a very interesting example, of course. – lulu Nov 20 '21 at 14:33
  • The book reads "it is to this cetain event that we assign a probability of 1". What do you think the book mean by the certain event? – Stats Cruncher Nov 20 '21 at 14:36
  • They say explicitly. The event they are referring to is "some event in the sample space occurs". – lulu Nov 20 '21 at 14:38
  • I am confused about that we assign a probability of 1 to the sample space not to an event. What is the difference? – Stats Cruncher Nov 20 '21 at 14:39
  • I think we are going in circles. As you must be aware, the probability of a set of events is defined to be the probability that at least one of the events in that set occurs. – lulu Nov 20 '21 at 14:41
  • If you toss an ordinary coin, the possible outcomes are Heads and Tails. Those two events comprise the sample space. Neither event is certain but it is certain that one of them will occur. – lulu Nov 20 '21 at 14:42
  • Thanks. the word one in "one of the possibilities in S must occur" means some instead of only one, is that what you mean? – Stats Cruncher Nov 20 '21 at 14:43
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    @lulu: I believe the usual terminology in probability is that the elements of the sample space are "outcomes" and and "event" means a set of outcomes. In particular the sample space itself is an event. One does not usually speak of "probability of a set of events". – Troposphere Nov 20 '21 at 14:43
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    Saying that one of the events must occur does not preclude the possibility that other events occur. If you want to do that, you should say "exactly one" or something to that effect. – lulu Nov 20 '21 at 14:44
  • @Troposphere Thanks, "the sample space itself is an event". It seems match the postulates. – Stats Cruncher Nov 20 '21 at 14:45
  • @lulu Appreciate the discussion. – Stats Cruncher Nov 20 '21 at 14:48

1 Answers1

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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.)

ryang
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  • The terminoligy 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? – Stats Cruncher Nov 21 '21 at 02:59
  • @ ryang Thanks. I am reading it. I also asked another question relating POSTULATES. Wow. – Stats Cruncher Nov 21 '21 at 03:55
  • Sure, appreciate your contribution. – Stats Cruncher Nov 21 '21 at 04:05
  • Could we CORRECTLY say that either at least one or exactly one of THE 16 OUTCOMES in the sample space must occur"? – Stats Cruncher Nov 21 '21 at 04:24
  • Is 15 from the number of a set {BBBW,BBWB,BBWW,BWBB,BWBW,BWWB,BWWW,WBBB,WBBW,WBWB,WBWW,WWBB,WWBW,WWWB,WWWW}? – Stats Cruncher Nov 21 '21 at 10:08
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    @StatsCruncher Good question! An event is just a set of outcomes that are of particular interest; to construct each event, just independently go through each (of the $16$) outcomes, deciding whether include or exclude it; this explains why there are $2^{16}$ events in total; “the outcome $BBBB$ can eventuate via $2^{15}$ different events” because there remain $15$ outcomes to consider for inclusion; half of the $65536$ events will contain the outcome $BBBB$. – ryang Nov 21 '21 at 12:13
  • A passage from the textbook reads *elements are more primitive or more elementary kinds of outcomes*. Now I can distinguish the difference in outcomes which are terms you have used to explain. – Stats Cruncher Nov 21 '21 at 13:38
  • @StatsCruncher Lacking context, I've no idea what that sentence means. But I can (repeating myself) say that "events (sets) contain outcomes (elements), and elementary events are simply single-outcome events". – ryang Nov 21 '21 at 15:25
  • Right, fair enough! – Stats Cruncher Nov 22 '21 at 01:52