In probability theory and statistics, the binomial distribution is the discrete probability distribution which gives only two possible results in an experiment, either Success or Failure. For example, if we toss a coin, there could be only two possible outcomes: heads or tails, and if any test is taken, then there could be only two results: pass or fail. This distribution is also called a binomial probability distribution.
There are two parameters n and p used here in a binomial distribution. The variable ‘n’ states the number of times the experiment runs and the variable ‘p’ tells the probability of any one outcome. Suppose a die is thrown randomly 10 times, then the probability of getting 2 for anyone throw is ⅙. When you throw the dice 10 times, you have a binomial distribution of n = 10 and p = ⅙.
Learn the formula to calculate the two outcome distribution among multiple experiments along with solved examples here in this article.
Also, read:
Binomial Probability Distribution
In binomial probability distribution, the number of ‘Success’ in a sequence of n experiments, where each time a question is asked for yes-no, then the boolean-valued outcome is represented either with success/yes/true/one (probability p) or failure/no/false/zero (probability q = 1 − p). A single success/failure test is also called a Bernoulli trial or Bernoulli experiment, and a series of outcomes is called a Bernoulli process. For n = 1, i.e. a single experiment, the binomial distribution is a Bernoulli distribution. The binomial distribution is the base for the famous binomial test of statistical importance.
Negative Binomial Distribution
In probability theory and statistics, the number of successes in a series of independent and identically distributed Bernoulli trials before a particularised number of failures happens. It is termed as the negative binomial distribution. Here the number of failures is denoted by ‘r’. For instance, if we throw a dice and determine the occurrence of 1 as a failure and all non-1’s as successes. Now, if we throw a dice frequently until 1 appears the third time, i.e., r = three failures, then the probability distribution of the number of non-1s that arrived would be the negative binomial distribution.
Binomial Distribution Examples
As we already know, binomial distribution gives the possibility of a different set of outcomes. In real life, the concept is used for:
- Finding the quantity of raw and used materials while making a product.
- Taking a survey of positive and negative reviews from the public for any specific product or place.
- By using the YES/ NO survey, we can check whether the number of persons views the particular channel.
- To find the number of male and female employees in an organisation.
- The number of votes collected by a candidate in an election is counted based on 0 or 1 probability.
Binomial Distribution Formula
The binomial distribution formula is for any random variable X, given by;
| P(x:n,p) = nCx px (1-p)n-x
Or P(x:n,p) = nCx px (q)n-x |
Where,
n = the number of experiments
x = 0, 1, 2, 3, 4, …
p = Probability of Success in a single experiment
q = Probability of Failure in a single experiment = 1 – p
The binomial distribution formula can also be written in the form of n-Bernoulli trials, where nCx = n!/x!(n-x)!. Hence,
P(x:n,p) = n!/[x!(n-x)!].px.(q)n-x
Properties of Binomial Distribution
The properties of the binomial distribution are:
- There are two possible outcomes: true or false, success or failure, yes or no.
- There is ‘n’ number of independent trials or a fixed number of n times repeated trials.
- The probability of success or failure varies for each trial.
- Only the number of success is calculated out of n independent trials.
- Every trial is an independent trial, which means the outcome of one trial does not affect the outcome of another trial.
Binomial Distribution Examples And Solutions
Example 1: If a coin is tossed 5 times, find the probability of:
(a) Exactly 2 heads
(b) At least 4 heads.
Solution:
(a) The repeated tossing of the coin is an example of a Bernoulli trial. According to the problem:
Number of trials: n=5
Probability of head: p= 1/2 and hence the probability of tail, q =1/2
For exactly two heads:
x=2
P(x=2) = 5C2 p2 q5-2 = 5! / 2! 3! × (½)2× (½)3
P(x=2) = 5/16
(b) For at least four heads,
x ≥ 4, P(x ≥ 4) = P(x = 4) + P(x=5)
Hence,
P(x = 4) = 5C4 p4 q5-4 = 5!/4! 1! × (½)4× (½)1 = 5/32
P(x = 5) = 5C5 p5 q5-5 = (½)5 = 1/32
Therefore,
P(x ≥ 4) = 5/32 + 1/32 = 6/32 = 3/16
Example 2: For the same question given above, find the probability of:
a) Getting at least 2 heads
Solution: P (at most 2 heads) = P(X ≤ 2) = P (X = 0) + P (X = 1)
P(X = 0) = (½)5 = 1/32
P(X=1) = 5C1 (½)5.= 5/32
Therefore,
P(X ≤ 2) = 1/32 + 5/32 = 3/16
Probability is a wide and very important topic for class 11 and class 12 students. By capturing the concepts here at BYJU’S, students can excel in the exams.