The binomial distribution is a representation of the results of repeated experiments which are independent and identical and in which there are two possible outcomes each time. This post constitutes Lesson 6 of the Basic Statistics Mini-Course.
You may also be interested in Bivariate statistics or two variable statistics.
Key concepts covered in this post: the binomial distribution, comparing the binomial distribution to the normal distribution, using the normal distribution to approximate the binomial distribution.
The binomial distribution
Example 1: Decide whether the following results can be represented by the binomial distribution:
1.1. Rolling 3 5s out of 15 rolls of a 6-sided die.
Yes, because each of the 15 rolls will yield only 2 (relevant) results, a 5 or not a 5. Each of the 15 rolls is identical and is independent of any other roll.
1.2. Rolling 3 5s or 3 2s out of 15 rolls of a 6-sided die.
No, because there are 3 possible outcomes on each of the 15 rolls (a 5, a 2, or another number).
The outcome of each repeated experiment is either a success or a failure.
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance. (Binomial distribution, 2021)
Example 2: Calculate the probabilities for success and failure of rolling 3 5s out of 15 rolls of a 6-sided die.
Assuming rolling a 5 as a success and not rolling a 5 as a failure, the probability of success on each trial is p = 1/6, while the probability of failure on each trial is q = 1 – 1/6 =5/6.
We can also work out the probability of a specific number of successes in a number of trials.
(Due to difficulties inserting some mathematical characters in WordPress, please finish reading this post in the embedded Google Doc below. You can jump to Example 3.)
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Bivariate statistics or two variable statistics
Data presentation in statistics
Normal distribution or Gaussian distribution
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