# Can a binomial experiment have 3 outcomes if they're treated as 2?

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?

• 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, $X'$. Jan 10, 2017 at 23:40
• 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. Jan 10, 2017 at 23:44

• 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. Jan 11, 2017 at 0:13