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A binomial experiment obviously needs to have only two possible outcomes. However, what if there are three outcomes but treated as two?

For example, would these two examples be binomial experiments?

  1. A researcher conducts a poll for 200 people in which the options are "Yes", "No", or "Maybe". X is the number of people that answer "Maybe".

  2. Someone spins a roulette wheel, which has green, black, and red spaces. X is the number of times the ball lands on a red space out of 20 trials.

It seems like there's only two outcomes for each. For the first example, the person either says "Maybe" or doesn't. For the second example, the ball lands on a red space or doesn't.

If they're not binomial experiments, why not?

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    $\begingroup$ These are binomial as you've defined them. In the first example you have defined an outcome $X$ and the other answers are simply grouped as the other outcome, $X'$. $\endgroup$
    – statsplease
    Commented Jan 10, 2017 at 23:40
  • $\begingroup$ I agree. It is just a matter of what you want to call a success whose probability p you wish to answer. But if you have three outcome and you are interested in p1 , p2 and p3 where p1+p2+p3=1 you can analyze the trinomial distribution. $\endgroup$
    – Michael R. Chernick
    Commented Jan 10, 2017 at 23:44

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Your intuition is correct. In fact, no matter how many outcomes an experiment has, you can always choose to group them together that there's only two outcomes, "X" and "not X", and then the experiment, as applied to X, will be a binomial experiment.

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    $\begingroup$ This is true but we should not neglect the fact that there is more information in the data than what you are using when you collapse categories and as I mentioned in my first remark, This is a little more complex, but you can analyze the multinomial distribution and estimate all k pi's. The question you are interested in should dictate the analysis. Why design the questionnaire to include the "maybe" response if you are only interested in "yes" or "no". Also do you throw out the maybes and just analyze the yes and nos? Then you sacrifice sample size. $\endgroup$
    – Michael R. Chernick
    Commented Jan 10, 2017 at 23:58
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    $\begingroup$ If you want to keep the original sample you can merge the maybes with one of the other categories. But should they go with the yeses or the nos? $\endgroup$
    – Michael R. Chernick
    Commented Jan 11, 2017 at 0:00
  • $\begingroup$ Of course, in the examples from OP, if the researcher is indeed interested in all three outcomes and their probabilities, then you can't use the binomial distribution. But the examples that OP gives for their variable $X$, it is correct that these examples are proper binomial experiments. $\endgroup$
    – Lagerbaer
    Commented Jan 11, 2017 at 0:13
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    $\begingroup$ Of course I agree with you. I was just trying to make the OP aware that he can analyze the data differently in case he is not aware of the possibility of using a multinomial. Note that at approximately the same time that you gave your answer I was saying the same thing in my comment. So we agree. $\endgroup$
    – Michael R. Chernick
    Commented Jan 11, 2017 at 0:51

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