Binomial Distribution

In probability and statistics theory, the binomial distribution can be defined as the probability which has a discrete nature and applies to events that have only 2 possible results within the experiment either failure or success. In binomial the prefix refers to twice or two. A few situations where instances of the binomial distribution are visible are the tossing of coins or rolling of dice. In coin tossing experiments, the tail or head can be taken as a failure or success. Such distribution involving a binomial random variable is known as the distribution of binomial probability. This distribution is generally used as a discrete distribution in statistics.

Binomial theorem formula

To understand the binomial distribution formula we first need to know the definition of the binomial distribution. The binomial distribution is described as the probability distribution of a particular binomial random variable. In this context, a random variable is defined as the real-valued function with the domain as the sample space within a particular random variable. Now, the binomial distribution formula can be discussed. The binomial distribution formula for any specific random variable Y is given by P(x: n,p) = nCx (1 – p)^{n – x} p^x or P(x: n,p) = nCx (q)^{n – x} p^x In the above formula, p is taken as success probability; the value is q is defined as the failure probability. Next n denotes the total number of trials. It is often seen that the binomial distribution formula is represented in the n – Bernoulli trial form. According to this form, nCx = n! / (n – x)! x!, and so P(x: n,p) = [n! / (n – x)! x!] . (q)n – x . px

Binomial random variable

To properly understand different aspects of a binomial random variable, one needs to have some prior knowledge concerning random variables as well as the different types of the random variable. A random variable is particularly that variable about whose value the user has no idea. It can also be regarded as a function that assigns value to the outcomes of a particular experiment. Random variables are two main types namely Continuous random variables and discrete random variables. Now the binomial random variable can be discussed. A binomial random variable is defined as the random variable which is used for counting the number of times a particular event occurs within a fixed number of trials. For instance, the total number of times heads are obtained when a coin is tossed 10 times is a binomial random variable. The conditions under which a random variable becomes a random variable have been outlined in the following.

•  Each trial must be classified as either failure or success.
• Trials should not depend on each other.
• The probability that a particular trial will succeed must be constant for each trial.
• The number of trials should be fixed.

Binomial distribution calculator

Many online sites provide a free binomial distribution calculator. One such online site is Giga calculator. This binomial distribution calculator can be used for calculating both the exact probability as well as the binomial probability of a particular distribution. To obtain the required probability one simply needs to enter the probability of success of a particular event, The total number of events for which the probability should be calculated, and the total number of trials. The calculator is applicable as long as the procedure for generating the events is conformable with the model of the random variable. In simple words, Y needs to be a random variable identically distributed binomially distributed and independent outcomes. For instance, one can easily calculate the probability of obtaining exactly five heads from a total of ten coin tosses. If the tossed coin is fair the probability obtained is 24.61 percent. Similarly, if one wants to find the probability that a particular fair dice will give two sixes when it is rolled twenty times then the required values will be entered in the binomial distribution calculator and the desired probability will be obtained as 67.13 percent.

Conclusion

In this article, the core topic that has been mostly analysed is Binomial distribution. This is an important statistical concept that has wide applications in the subject of Agricultural Engineering. To analyse this topic first, a binomial distribution formula has been identified. Next, discussions have been done on a binomial random variable which has been followed by proper assessment of the binomial distribution calculator.