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A binomial distribution is defined as a probability distribution defined on a discrete set of random variables that describes the number of predefined successes in an experiment. Since the binomial distribution is defined for a binary experiment, its probability can be described as a binomial expansion where each term represents the probability of a specific number of successes out of n times the experiment is repeated.
Thus, the binomial random variables is defined as the number of successes in an experiment repeated n time. Binomial distributions are the basics in the study of probability distribution which is one of the most important concepts in applicationbased Statistics.
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A binomial random variable is a variable that represents the number of successes in an ntimes repeated binary experiment (outcomes are either success or failure) and is known as the binomial random variable.
A binomial distribution is described as a probability distribution based on a discrete set of random variables, the symbolic representation of a Binomial distribution, where n represents the number of times the experiment has been repeated, and p is the probability of success in a single iteration of the experiment. A binomial variable is the probability of a random variable in a binomial distribution.
So, the relationship between binomial variables and binomial random variables is that a binomial variable is a specific outcome of a binomial random variable, which is a random variable that follows a binomial distribution
The binomial distribution is different from other distributions in certain aspects. These are known as the characteristics of binomial distribution or binomial variables and they are given as follows,
Some distinct properties of binomial distribution distinguish them from other probability distributions. These properties are as follows.
And the standard deviation is given by,
Binomial Distribution is very useful in many reallife practices, we find the applications of binomial distribution in various situations and mathematical problems some of the applications are given below,
Binomial distribution has many different terms and identifiers which can be used to determine if a given distribution is a binomial distribution or not and to calculate probabilities for different random variables.
In this article, we have learned about binomial distributions and binomial random variables. A binomial distribution is a probability distribution defined on a finite discrete set of random variables, for a fixed number of trials of a binary experiment, i.e., there are only 2 possible outcomes to one trial of the experiment, either success or failure.
A binomial variable function is the probability of a random variable in a binomial distribution. When represented as a probability distribution table, binomial distribution can be used to calculate expected value, variance, and standard deviation random variable.
Now let’s see some questions based on Binomial Probability distributions.
Example 1: Find the probability of 5 heads in 7 tosses of a coin.
Solution: Here in this question, we have 7 total coin tosses, so the number of trials is a fixed finite number, i.e., 7.
Also, a coin toss has only two outcomes and we are given that we need to find the probability of heads, so we can consider getting a head on a toss as our success, then we have
And then, the probability of failure would be,
Hence, the probability of success and failure is a constant independent of the number of trials or random variable, thus the given question can be represented by the following Binomial Variable,
To calculate,
In this calculation, the term
is called a binomial coefficient and it is pronounced as 7 choose 5, and the formula for any n choose r, , is given by
Then we have,
Hence, the probability of getting 5 Heads in 7 coin tosses is 21 out of 128.
Example 2: Find the Expected value and Standard Deviation, in a binomial distribution, when the total number of trials is given to be 5 and the probability of success is .
Solution: Let’s make a Probability distribution table from the given Binomial Distribution





0 
 0  0  0 
1 
 1  
2 
 4  
3 
 9  
4 
 16  
5 
 25  
The mean/Expected value is given by,
Variance is given by,
And the standard deviation,
Hence the Expected value is 1.875 and the variance is 3.125.
Example 3: Find the probability of getting a composite number at least three times on 4 throws of an unbiased dice.
Solution: Here we have, 4 throws of an unbiased dice, i.e., the number of trials is a fixed finite number. Next, we need to calculate the probability of getting a composite number, so we can define success in a trial as getting a composite number. Hence, we have
(We have 2 composite numbers, 4 and 6, out of 6 total outcomes)
And the corresponding failure is,
Since we need to find the probability of at least 3 composite numbers, then we have the random variable, .
Then calculate the probability of 3 composite numbers,
And the probability of 4 composite numbers,
And finally,
Thus, the probability of at least 3 composite numbers in 4 throws of unbiased dice is 1 out of 9.
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Edwards, A. W. F. (1960). The meaning of binomial distribution. Nature, 186(4730), 10741074.