Binomial Distribution Explained: What Is Binomial Distribution?

A binomial distribution is a probability distribution that shows the probability “P” of “n” trials, where each trial has two possible outcomes. The value “n” can be any positive integer. That includes the number one, which means you can do a probability distribution for a single trial.

A simple way to visualize binomial probability is to think of a coin flip. There are only two possible values in a fair coin toss: heads or tails. If you call one of these outcomes a "success," then you can calculate the probability of success in each coin flip and the cumulative probability of success in multiple coin flips. If you say that "heads" is a success, that means that the probability of success is the probability of the coin landing on heads. Accordingly, you would call landing on "tails" a failure (in the language of probability). Therefore, the probability of failure is the probability of landing on tails.

You must use a series of calculations to determine the probability of X successes in n number of trials when each trial has two possible outcomes. For example, if you are predicting coin tosses, the number of successes would refer to the total number of heads or the total number of tails that come up as you continually toss a coin. To compute binomial distributions, you will need some knowledge of factorials and order of operations. The binomial formula is:

b(x; n, P) = { n! / [ x! (n - x)! ] } * Px * (1 - P)n - x

• “P” is probability
• “n” is the number of trials
• “x” is the number of successes
• “b(x; n, P)” means the binomial probability of x successes in n trials

Using data from a binomial distribution, you can calculate the expected values of a random variable as it goes through independent trials. In the case of the classic binomial outcome—the results of a coin toss—you can calculate the probability of getting an exact number of successes in an exact number of trials. In other words, you can predict the exact number of heads or number of tails you should expect when you flip a coin a certain number of times.

You can also use cumulative binomial probability to find the probability of getting a certain range of outcomes. For instance, you could decide you want to know the probability of flipping a coin one hundred times and getting heads thirty times or fewer. You would use a more complex variation on the binomial formula to find that probability. You would input a sample size of one hundred and an outcome range of zero to thirty and then run a long string of calculations to arrive at the needed probability.

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