I just got my first assignment in probability back with 23/25 marks. The two-point deduction was for my answer to the following question. I know I am wrong (after all, to think otherwise would be refuting the professor!). However, it's very frustrating because it's not really a case in which I could have known the math better. My thought processes and interpretation of the problem were such that I never stood a chance of getting it right. Those are the kinds of mistakes that are hard to learn from, so I need your help:(adsbygoogle = window.adsbygoogle || []).push({});

A die is rolled continually until a 6 appears, at which point the experiment stops. What is the sample space of the experiment? Let E_{n}denote the event that n rolls are necessary to complete the experiment. What points of the sample space are contained in E_{n}? What is [itex] (\bigcup^{\infty}_{n=1} {E_n})^c [/itex]?

superscript c denotes the complement of the event.

My solution, just for the record:

X Wrong! The sample space, S, is the set of all possible outcomes, which is the set of all possible numbers, "n", of rolls required before a six is rolled:

[tex] S = \{n| \ \ 1 \leq n < \infty, \ \ n \in \mathbb{N} \} [/tex]

The event E_{n}has only one point in the sample space; it is the outcome that n rolls are required to roll a six. The union of the E_{n}'s makes up the entire sample spacebecause it is always possible to roll a six in some finite number of n rolls in the interval[1, [itex] \infty [/itex])

Therefore:

[tex] \bigcup^{\infty}_{n=1} {E_n} = S [/tex]

and

[tex] (\bigcup^{\infty}_{n=1} {E_n})^c = {\O} [/tex]

(note, I am not capable of remembering conversations verbatim. I have reproduced here the essence of my discussion with the prof after class:)

When I question my prof today, he said: "You didn't include infinity."

"Huh?"

"You didn't include infinity in your sample space." I thought about it for a second and replied, "You mean the outcome that a 6 is never rolled?"

"Yes," he confirmed.

"But, but," I sputtered, "you proved later in class quite generally that in experiments such as these, it is always possible to reach the desired outcome in some finite number of trials (ie in this instance, P(never roll a six) = 0)." In the back of my mind, I was thinking, that even though he hadn't yet proved it at the time I wrote the assignment, I thought it was so trivially obvious, that I had assumed it to be true in my solution (see italicized statement). Was he saying this assumption was wrong?

"Sure," he said. "I proved that. But this question makes no reference to probability whatsoever. You're just thinking about what outcomes ought to be in the sample space. The union that was given in the question is a never ending sequence of events in which ever larger numbers of rolls 'n' occur before a six is reached. However, this union does not include [itex] E_{\infty} [/itex] itself. Therefore: "

[tex] (\bigcup^{\infty}_{n=1} {E_n})^c = \{\infty\} [/tex]

So he wasn't saying that the italics statement was wrong, but that it was irrelevant to the question. I was so befuddled that I couldn't come up with a satisfactory response. If I had had my wits about me, my retort would have been: "But the sample space is the set of allpossibleoutcomes, so if it isimpossiblenever to roll a six, then why the heck is that outcome in the sample space?" As it was, it was time to leave, so the discussion ended there. As we were leaving class, my friend tried to make sense of it with me. He reasoned that "the set of possible outcomes" was determined simply by the way an "outcome" was defined by the experiment, not by probability, which we were not considering in the question. To me, however, this is a question of semantics. "Possible" should mean that there is a chance of it actually happening in reality, should it not? Any clarification would be greatly appreciated.

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# A Question of Semantics?

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