Should the Possibility of Never Rolling a Six be Included in the Sample Space?

[cepheid]

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:

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 $(\bigcup^{\infty}_{n=1} {E_n})^c$?

superscript c denotes the complement of the event.

My solution, just for the record:

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:

$$S = \{n| \ \ 1 \leq n < \infty, \ \ n \in \mathbb{N} \}$$

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 space because it is always possible to roll a six in some finite number of n rolls in the interval [1, $\infty$)

Therefore:

$$\bigcup^{\infty}_{n=1} {E_n} = S$$

and
$$(\bigcup^{\infty}_{n=1} {E_n})^c = {\O}$$

X Wrong!

(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 $E_{\infty}$ itself. Therefore: "

$$(\bigcup^{\infty}_{n=1} {E_n})^c = \{\infty\}$$

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 all possible outcomes, so if it is impossible never 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.

 

[HallsofIvy]

"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 "

Your professor surely did NOT prove that (or if he did his proof was wrong) because it is not true. It is quite possible, in fact, to roll a "1" repeatedly, never getting any other value. Of course, the probability of rolling n "1"s is (1/6)ⁿ so that this goes to 0 very fast.

He/she may well have have proved that the probability of an infinite string that does NOT contain a "6" is 0, or, conversely, that the probability that the experiment will terminate after a finite number of rolls is 1, but that is NOT the same as saying one is "impossible" or the other is "certain".

Yes, it is true that the probability of rolling an infinite string of "1"s (i.e. always getting a "1") is 0, or that the probability that an infinite string of rolls does NOT include a "6" is 0. However, in dealing probability with an infinite number of possible outcomes, a probability of 0 does NOT mean it is impossible. Whatever string of rolls you DO get, rolling an infinite number of dice, must also have probability 0 but it DID happen!

If you choose a number between 0 and 1 with uniform probability, the probability of getting ANY specific number is 0 but SOME number must be chosen.

 

[tacman]

It sounds like you had a frustrating experience with this question, but it's great that you are seeking clarification and trying to understand the reasoning behind the correct answer.

In this case, it seems like the issue is indeed a semantic one. When we talk about the sample space in probability, we are referring to the set of all possible outcomes of an experiment, regardless of whether they are likely or not. So in this case, the sample space does include the outcome of never rolling a six, even though it is impossible in reality.

The key here is that the sample space is determined by the experiment itself, not by probability. The experiment allows for the possibility of never rolling a six, even though it is not a likely outcome. So in this context, "possible" means that it is a potential outcome within the framework of the experiment, not necessarily in reality.

I hope this explanation helps clarify things for you. It's important to keep in mind the specific context and definitions when dealing with probability and sample spaces. Keep up the good work and don't get discouraged by this experience!