Binomial distribution

Introduction

In statistics as well as probability theory, the binomial distribution is referred to as the probability distribution which provides only two possible outcomes within an experiment which is either failure or success. For instance, if a coin is tossed there can be the possible outcomes of tails or heads. Further, if any test is taken then only two possible results can be obtained that fail or pass. This distribution is also known as a binomial probability distribution. 

Binomial distribution Formula

In a Binomial distribution of probability, the number of times the trial will be successful in a series of n experiments in which whenever a question is asked in the form of a yes or no, then the Boolean valued outcome is shown either in the form of true/ one/ yes/ success (probability p) or no / zero/ false/ failure (probability q or 1 – p).  

The formula for binomial distribution is used for any variable X that is random in nature is given by 

P (x:n, p) =  Cxn* px (1-p)n-x  or P (x:n, p) = Cxn * px (q)n-x 

Where n is denoted as the number of times the experiment has been done. prefers to the probability of success within a particular experiment. q is 1-p which denotes the probability of failing within a particular experiment.

The formula for binomial distribution can be further written as n – Bernoulli trials. In this form Crn is taken as n! / x! (n-x)!

Thus the overall formula becomes P(x:n, p) = n! / {x! (n-x)!} . px (q)n-x

Binomial distribution Calculator

A binomial distribution calculator is a tool that can be used for calculating any probability of a binomial distribution. This tool can either be used for the calculation of cumulative probability for observing X ≥ x or the cumulative X ≤ x or X > x or X < x. For this simply the probability needs to be inserted through observation of a particular event that is the outcome of success or interest within a single trial, the number of events, and the number of trials that an individual needs to calculate the probability for. The binomial distribution calculator is conformable as long as the model for a random variable is confirmed by the procedure of generation of an event under the binomial distribution. In simple terms, X needs to be a random variable whose generation is a result of a particular method that results in identical, binomially distributed, and independently distributed outcomes. For instance, a person can calculate the probability such that exactly 5 heads are obtained from tossing a fair coin 10 times is 24.61 percent. Also, the probability of obtaining 2 sixes within a series of rolling a dice 20 times is 67.13 percent. 

Binomial distribution Example

Example 1: After tossing a coin 5 times, calculate the probability such that there are exactly (1) two heads exactly, and (2) At least four heads. 

Answer: Repeatedly tossing one coin is an instance of the Bernoulli trial. From the given problem we get that the Total trial number is 5. The probability such that a head will occur is p =1/2 and for this reason, the probability that a tail will occur is also q = ½. 

1. Now to obtain exactly two heads x = 2.

Therefore P(x= 2) is C25 p2 q5 – 2 = 5! / 3! 2! * (1/2)2 * (1/2)3 = 5 / 16.

2. To obtain at least four heads, x ≥ 4 and so P (x ≥ 4) = P (x = 5) + P (x = 4) 

Therefore, P(x= 4) is C45 p4 q5 – 4 = 5! / 1! 4! * (1/2)4 * (1/2)1 = 5 / 32.

P(x= 5) is C55 p5 q5 – 5 = 5! / 3! 2! * (1/2)5 = 1 / 32.

and so, P (x ≥ 4) is 1/32 + 5/32 = 6 / 32 which implies 3/16.

Conclusion 

The main subject on which the article has been written is Statistics. Under this, the main topic that has been analyzed throughout the research is the binomial distribution. From the overall discussion, it has been identified that binomial distribution is a type of discrete distribution. Under the main topic of the binomial distribution, a number of subtopics have been discussed. These involve binomial distribution formulas, binomial distribution calculators, and binomial distribution examples.