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Say I have the problem: I roll a die three times and I am interested in the probability of ending up with two 1's.

My impression is that a single roll is called a trial.

What is the full 3-roll action called? I thought it was called an experiment, but I've seen experiment used synonymous with trial.

Using the binomial distribution, we call getting a 1 here a "success" and getting anything else a "failure." This might confuse students who think it's a "failure" to get three 1's. Is there an alternative/better way to say this acquiring of a 1 vs. not a 1?

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    $\begingroup$ "ending up with two 1's" Do you mean exactly two or at least two? $\endgroup$
    – JRN
    Apr 12, 2021 at 0:45
  • $\begingroup$ I mean exactly two 1's. $\endgroup$
    – Nights
    Apr 12, 2021 at 8:22
  • $\begingroup$ You should edit your post to reflect that. $\endgroup$
    – JRN
    Apr 12, 2021 at 8:51

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  • You might point out that the $3$-trial experiment has $8$ possible outcomes and $256$ possible events, which are subsets of the sample space; whereas each trial has $2$ possible outcomes, which can be contextually characterised as a success and a failure.

    To distinguish between trial outcomes and experiment outcomes (or to remind about their distinction), I sometimes call the former “sub-outcomes” and the latter “elementary events”. When introducing the terminology, I indicate that “sub-outcome” is a nonstandard word (actually, I coined it).

  • To directly address your question:

    I would point out that the assignments ‘success’ & ‘failure’ naturally arise out of Bernoulli trials having exactly two outcomes of fixed probabilities.

    In contrast, it is not so useful to call the event $\{n11,1n1,11n\}$ or its outcomes “successes”; moreover, we might be interested in multiple events in the experiment, in which case the notion of a “successful” event (or elementary event) will have to keep varying according to the event being considered.

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  • $\begingroup$ So you agree that experiment is the word for the full 3-roll activity, and is not synonymous with trial? $\endgroup$
    – Nights
    Apr 12, 2021 at 8:25
  • $\begingroup$ Apparently my stats are rusty because I don't know where you're getting 256. Do you mean 216 (6*6*6)? $\endgroup$
    – Nights
    Apr 12, 2021 at 8:31
  • $\begingroup$ @Nights 1. Yes. And given enough discussion time, I might also point out that while any two trials of this experiment are independent of each other, two events with reference to this experiment could be either dependent or independent. $\quad$2. The sample space has $2^3=8$ members/outcomes and thus $2^8=256$ subsets/events. $\endgroup$
    – ryang
    Apr 12, 2021 at 8:39
  • $\begingroup$ There are six outcomes in a trial: 1 - 6. When you write out every possibility you have 216 possible 3-die rolls. I thought that THIS constitutes the sample space (I haven't opened the class textbook yet, but this is what I remember.) The probability is calculated by hand by finding the number of rolls which give two 1's, where a roll (1,1,2) is different from a roll of (1,1,3), for example, and dividing this number by 216. Where are you getting this other use of the term "sample space" in terms of success/failure only? And what's the significance of your power set calculation? $\endgroup$
    – Nights
    Apr 12, 2021 at 9:16
  • $\begingroup$ 1. In your binomial model, each trial has exactly two possible outcomes: ‘1’ (success) and ‘n’ (failure, whose likelihood here is five times that of success). The game can of course be reframed as a different probability experiment in which each trial has six equally-likely outcomes; in this model, the experiment indeed has $6^3=216$ outcomes. $\endgroup$
    – ryang
    Apr 12, 2021 at 10:11

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