The Swiss mathematician James Bernoulli (1654–1705), created the binomial distribution in 1700, and it was initially published in 1713, eight years after his passing. The following circumstances allow for the usage of this distribution:

- The random experiment is repeated a set number of times.
- The results of each trial can be divided into two mutually exclusive groups named success (the occurrence of the event) and failure (the non-occurrence of the event).
- Every trial was conducted independently.
- For each trial, the chance that success (the occurrence of an event) will occur remains constant at p.

For instance, if we flip a fair coin n times, the result of any trial will be one of the two mutually exclusive outcomes, namely head (success) or tail (failure). Additionally, each trial is independent of the others because the outcome of any given coin toss has no bearing on or influence over the outcomes of the others. Additionally, the chance of success (head) in any trial is always half, making it a constant throughout all trials. Thus, the Binomial distribution will emerge from the coin-tossing problems. Problems involving the roll of the dice will also follow the Binomial distribution.

**Binomial Distribution Calculator:**

The Binomial Distribution Calculator below calculates the probability that a binomial random variable takes a value equal to the given value of x. It also calculates the probabilities of taking value less than x, less than equal to x, more than x, and more than equal to x.

P(X=43) = 0.03007

P(X<43) = 0.06661

P(X≤43) = 0.09667

P(X>43) = 0.90333

P(X≥43) = 0.93339