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# A Detailed Explanation of the Binomial Probability Formula

A binomial distribution is simply the probability of success or failure in a conducted trial that is repeated multiple times. When we repeat a set of events say tossing a coin for 10 times, each trail in the set has two possible outcomes I.e. head or tails. Every single trail is known as a Bernoulli […]

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A binomial distribution is simply the probability of success or failure in a conducted trial that is repeated multiple times. When we repeat a set of events say tossing a coin for 10 times, each trail in the set has two possible outcomes I.e. head or tails. Every single trail is known as a Bernoulli trial. The term Binomial distribution indicates bi i.e. two outcomes define as success or failure.

The binomial probability formula is used to interpret future events. The binomial distribution functions are often used to predict the result of any real-life events that satisfy the assumption of the method. Considering the importance of binomial probability in real life statistical problems we are here providing you all the basic of binomial distribution:

## What is binomial distribution:

As explained before the result of an experiment will always have only two possible outcomes in the case of a binomial distribution. Hence, the 2 parameters to address the binomial condition are:

“P” which refers to the probability of any one event.

“n” which refers to the numbers of time the trails have been conducted

The binomial concept can also be explained by “Boolean-valued outcome”. the result of an experiment can be in yes or no. According to the Boolean-value, success can be represented as true/success/yes/1. It is the binomial distribution for value P whereas the failure can be denoted in the form false/failure/no and represent as q = 1 – P.

Some of the important definitions you need to know about the binomial probability formula are:

Bernoulli trial: it can be described as an experiment where only one outcome result is possible out of two I.e either failure or success.

Binomial distribution: the representation of a single condition experiment is described by the binomial distribution. Where n= 1.

Bernoulli process: when there are more than two trial outcomes, the sequencing criteria of all the results is referred to as a Bernoulli process.

## Binomial probability formula:

Taking any random variable, the binomial probability formula can be represented as:

P(x:n,p) = nCx px (1-p)n-x OR nCx px (q)n-x

For n– Bernoulli trials the formula can be written as:

P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x

Here:
X is the variables 1, 2, 3, 4, 5, and so on.
‘n’ represents the total no. of trials
‘q’ is the total failure probability of a single trial for 1-p
‘p’ is the probability rate of success

## Negative binomial distribution:

In the case of the binomial distribution, the conducted trails of an experiment are identical yet independent of each other. Before counting the false “r”, the total no. of success that happened first is described as a negative binomial probability.

For example, you are tossing two coins at once. Say getting 2 head on both the coin is success and all the other outcomes i.e. head and tail or tail and tail are a failure. Here the pair of coins is tossed for 10 times, before getting head on both the coins. Here 10 (r) is the condition (the rate of failure) when we didn’t get the 2 heads. Hence, the tosses that didn’t have 2 heads initially is the negative binomial probability distribution.