In an experiment or survey that is replicated several times, a **binomial distribution** can be thought of as simply the likelihood of a SUCCESS or FAILURE result. It is a probability distribution that summarizes the probability that, under a given set of parameters or assumptions, a value will take on one of two independent values. When each trial has the same probability of achieving one particular result, it summarizes the number of trials. Multiplying the number of independent trials by the achievements obtains the value of a binomial.

In a series of n independent experiments, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes, each asking a yes-no question, and each with its own Boolean-valued result: success / yes / true / one (with probability p) or failure / no / false / zero (with probability q= 1 − p). The binomial is a distribution form with two potential outcomes (the “bi” prefix means two, or twice). A coin flip, for example, only has two possible results: heads or tails, and taking a test may have two possible results: pass or fail. A single success/failure experiment is also referred to as a Bernoulli or Bernoulli experiment, and a series of results is called a Bernoulli process; the binomial distribution is a Bernoulli distribution for a single trial, i.e. n = 1.

**Example of Binomial Distribution**

The fundamental suppositions of the binomial distribution are that there is just a single result for every preliminary, that every preliminary has a similar likelihood of accomplishment, and that every preliminary is totally unrelated, or autonomous of one another. Binomial distribution models the likelihood of event of an occasion when explicit measures are met. If each Bernoulli trial is independent, then the number of successes on Bernoulli trails has a binomial distribution, according to Washington State University. The Bernoulli distribution, on the other hand, is the Binomial distribution of n=1.

Binomial distribution sums up the quantity of preliminaries, or perceptions when every preliminary has a similar likelihood of achieving one specific worth. The binomial conveyance decides the likelihood of watching a predefined number of effective results in a predetermined number of preliminaries. In every preliminary, the likelihood of accomplishment, P(S) = p, is the equivalent. Just 1 minus the probability of success is the probability of failure: P(F) = 1-p. In social science statistics, the binomial distribution is also used as a building block for models of dichotomous outcome variables, such as whether a Republican or Democrat will win an upcoming election or whether a person will die within a certain period of time, etc.

The binomial distribution formula is:

**b(x; n, P) = _{n}C_{x} * P^{x} * (1 – P)^{n – x}**

Where:

**b** = binomial probability

**x** = total number of “successes” (pass or fail, heads or tails, etc.)

**P** = probability of success on an individual trial

**n** = number of trials

**Example of Binomial Distribution:** Suppose, according to the latest police statistics, 80% of all petty crimes are unsolved, and at least three of those are committed in our city. Both of the three offenses are independent of one another.

**Solution:**

The first step in finding the likelihood of the binomial is to check that the situation follows the four binomial distribution laws:

- Number of fixed trials (n): 3 (Number of petty crimes)
- Number of mutually exclusive outcomes: 2 (solved and unsolved)
- The probability of success (p): 0.2 (20% of cases are solved)
- Independent trials: Yes

**Next:**

In the three separate trials, we see the possibility that one of the crimes will be solved. It is shown as follows:

**Trial 1** = Solved 1st, unsolved 2nd, and unsolved 3rd

= 0.2 x 0. 8 x 0.8

= 0.128

**Trial 2** = Unsolved 1st, solved 2nd, and unsolved 3rd

= 0.8 x 0.2 x 0.8

= 0.128

**Trial 3** = Unsolved 1st, unsolved 2nd, and solved 3rd

= 0.8 x 0.8 x 0.2

= 0.128

Total (for the three trials):

= 0.128 + 0.128 + 0.128

= 0.384

Alternatively, in the binomial probability formula, we may apply the knowledge as follows:

Where,

**n / x = n! / x!(n-x)!**

In the equation, x = 1 and n = 3. The equation gives a probability of 0.384.

In real life, several examples of binomial distributions can be found. It is the sum of a sequence of several Bernoulli trials, separate and identically distributed. The experiment is said to be random and can only have two potential results in a Bernoulli trial: success or failure.

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