What Is a Binomial Distribution?.
A probability distribution is a function or rule that assigns probabilities of occurrence to each possible outcome of a random event. Probability distributions give us a visual representation of all possible outcomes of some event and the likelihood of obtaining one outcome relative to the other possible outcomes.
A binomial distribution is a specific probability distribution. It is used to model the probability of obtaining one of two outcomes, a certain number of times (k), out of fixed number of trials (N) of a discrete random event.
A binomial distribution has only two outcomes: the expected outcome is called a success and any other outcome is a failure. The probability of a successful outcome is p and the probability of a failure is 1 - p.
A successful outcome doesn't mean that it's a favorable outcome, but just the outcome being counted. Let's say a discrete random event was the number of persons shot by firearms last year. We'd be looking for the probability of obtaining some number of victims out of the pool of shootings. Being shot is neither a favorable nor a successful outcome for the victim, yet it is the outcome we are counting for this discrete variable.
Criteria for Using Binomial Distributions
The binomial distribution is used to model the probabilities of occurrences when specific rules are met.
- Rule #1: There are only two mutually exclusive outcomes for a discrete random variable (i.e., success or failure).
- Rule #2: There is a fixed number of repeated trials (i.e., successive tests with no outcome excluded).
- Rule #3: Each trial is an independent event (meaning the result of one trial doesn't affect the results of subsequent trials).
- Rule #4: The probability of success for each trial is fixed (i.e., the probability of obtaining a successful outcome is the same for all trials).
When a Successful Outcome is an Inequality
What about when the successful outcome is not exactly one outcome? Let's rewrite the first situation as this:
Given a couple has 5 children, what is the probability that 3 or more will be boys?
Possible outcomes: boy or girl
Fixed number of repeated independent trials: 5
Out of 5 trials, either 3, 4, or 5 are boys = success
Probability of success (0.5) + probability of failure (0.5) = 1.
When the successful outcome takes on more than one exact value, then we are looking for the probability of a cumulative binomial distribution. Cumulative binomial distributions are calculated differently than when successes can take on only one value. The binomial distribution formula applies to situations that do not include cumulative probabilities.