How to Tell When a Random Variable Doesn’t Have a Binomial Distribution
In order to know when a random variable in a statistical sample does not have a binomial distribution, you first have to know what makes it binomial. You can identify a random variable as being binomial if the following four conditions are met:
There are a fixed number of trials (n).
Each trial has two possible outcomes: success or failure.
The probability of success (call it p) is the same for each trial.
The trials are independent, meaning the outcome of one trial doesn’t influence that of any other.
So if it doesn’t meet all of these conditions, you can say that a random variable is not binomial.
Distribution is not binomial when the number of trials can change
Suppose that you’re going to flip a fair coin until you get four heads and you’ll count how many flips it takes to get there; in this case X = number of flips. This certainly sounds like a binomial situation: Condition 2 is met because you have success (heads) and failure (tails) on each flip; Condition 3 is met with the probability of success (heads) being the same (0.5) on each flip; and the flips are independent, so Condition 4 is met.
However, notice that X isn’t counting the number of heads (successes), it counts the number of flips (trials) needed to get 4 heads. The number of successes (X) is fixed rather than the number of trials (n). Since the number of trials is not fixed, Condition 1 is not met, so X does not have a binomial distribution in this case.
Distribution is not binomial when there are more than two outcomes
Some situations involve more than two possible outcomes, yet they can appear to be binomial. For example, suppose you roll a fair die 10 times and let X be the outcome of each roll (1, 2, 3, . . . , 6). You have a series of n = 10 trials, they are independent, and the probability of each outcome is the same for each roll. However, on each roll you’re recording the outcome on a six-sided die, a number from 1 to 6. This is not a success/failure situation, so Condition 2 is not met.
However, depending on what you’re recording, situations originally having more than two outcomes can fall under the binomial category. For example, if you roll a fair die 10 times and each time you record whether or not you get a 1, then Condition 2 is met because your two outcomes of interest are getting a 1 (“success”) and not getting a 1 (“failure”). In this case, p (the probability of success) = 1/6, and 5/6 is the probability of failure. So if X is counting the number of 1s you get in 10 rolls, X is a binomial random variable.
Distribution is not binomial when the trials aren’t independent
You have 10 people — 6 women and 4 men — and you want to form a committee of 2 people at random. Let X be the number of women on the committee of 2. The chance of selecting a woman at random on the first try is 6/10.
Because you can’t select this same woman again, the chance of selecting another woman is now 5/9. The value of p has changed, and Condition 3 is not met.
In this example, it is also the case that Condition 4 is not met. If the first person selected is a woman, then the chance of selecting another woman is 5/9. But if the first person selected is a man, then the chance of selecting a woman on the second try is 6/9. The outcome of the first try influences the outcome of the second try, thus, selections are not independent.
If the population is very large (for example all U.S. adults), p still changes every time you choose someone, but the change is negligible, so you don’t worry about it. You still say the trials are independent with the same probability of success, p. (Life is so much easier that way!)
Distribution is not binomial when the probability of success (p) changes
You have 5 urns: A, B, C, D, E. Urns A and B have balls numbered 1 through 5; urns C, D, E have ball numbers 1 through 10. There are five trials. In each trial, you draw a ball from an urn. In the first trial you draw from urn A, in the second trial you draw from urn B, etc. Let X be the number of times you draw a ball numbered 1.
This would not be a binomial distribution because the probability changes. In the first two trials (using urns A and B), the probability of success is 1/5. But in the next three trials (using urns C, D, and E), the probability of success is 1/10.