Properties
- A binomial distribution represents an experiment that consists of a fixed number of trials denoted as n.
- Each of the trials in the experiment has only two results - either a success or a failure, and each has a probability of success denoted as p.
- The trials are identical and independent.
- The random variable is the number of successes from the fixed number of trials. This random variable is denoted as x.
What do these properties mean?
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Fixed Number of Trials
This means that the number of trials that are going to be conducted in the experiment are determined beforehand. Once the experiment has begun, this number will not change.
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Success or Failure
The definition of a success in a binomial experiment is that you got whatever you were looking for. Therefore, the definition of a failure in a binomial experiment is that you did not get whatever you were looking for.
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Identical and Independent
The trials being identical and independent means that the result of one trial has no effect on any other trials. This means that the probability of a success does not change trial to trial.
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Random Variable
The random variable is the thing in the experiment that is not determined beforehand - it is likely to change from experiment to experiment. In this case, the number of successes is the random variable, meaning that the number of successes from the experiment is unknown until the experiment is over.
Example
When dealing with an experiment, if you see or hear the words "with replacement" you are probably dealing with a binomial experiment.
For example, imagine you have a standard deck of 52 cards and you are interested in seeing how many draws out of 10 are a face card (Jacks, Queens, and Kings). 10 times you will take a single card out of the deck, check if it is a face card, and then immediately replace it in the deck. This would be a binomial distribution with n = 10 (because there are 10 trials), and p = 12/52 or about 0.2308 (because there are 3 face cards and 4 suits of each). The only thing left would be to determine how many of those 10 cards you would like to be face cards, and that would be x (because it is the number of successes you would like). Note that you will not always get exactly that number of successes - there is only some probability that you will.
Requirements
- The number of trials must be a whole number at least equal to the number of successes you are looking for and must also be greater than zero (n >= x and n > 0).
- The number of successes must be a whole number greater than or equal to zero (x = 0, 1, 2 ... n).
- The probability of success must be greater than or equal to zero and also less than or equal to one (1 >= p >= 0).