Binomial Distribution in Probability
Last Updated :
04 Dec, 2024
Binomial Distribution is a probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is useful for calculating the probability of a specific number of successes in scenarios like flipping coins, quality control, or survey predictions.
Binomial Distribution is based on Bernoulli trials, where each trial has an independent and identical chance of success. The probability distribution for a Bernoulli trial is called the Bernoulli Distribution.
A Binomial Distribution for a random variable X = 0, 1, 2,…, n is defined as the probability of success or failure in a series of independent trials. Each trial is independent of the others, and the distribution helps calculate the probability of various outcomes in these trials.
Conditions for Binomial Distribution
The Binomial distribution can be used in scenarios where the following conditions are satisfied:
- Fixed Number of Trials: There are a set number of trials or experiments (denoted by n), such as flipping a coin 10 times.
- Two Possible Outcomes: Each trial has only two possible outcomes, often labeled as "success" and "failure." For example, getting heads or tails in a coin flip.
- Independent Trials: The outcome of each trial is independent of the others, meaning the result of one trial does not affect the result of another.
- Constant Probability: The probability of success (denoted by p) remains the same for each trial. For example, if you’re flipping a fair coin, the probability of getting heads is always 0.5.
The Binomial distribution is an appropriate model to use for calculating the probabilities of obtaining a certain number of successes in the given trials.
Read More: Bernoulli trails
Negative Binomial Distribution
The Negative Binomial Distribution is used to model the number of trials needed to achieve a certain number of successes in a sequence of independent trials, where the probability of success in each trial is constant.
For example, consider a situation where getting 6 is the success of throwing a die. Now if we throw the die and not get 6 then it is a failure. Now we throw again and do not get 6. Let's say we don't get 6 for three successive attempts and 6 is obtained in the fourth attempt and onwards then the binomial distribution of the number of getting 6 is called the Negative Binomial Distribution.
The formula for Negative Binomial Distribution is given as
P(x) = n+r-1Cr-1prqn
Where,
- n = Total Number of Trials.
- r = Number of Trials in which we get the first success.
- p = Probability of Success in Each Trial.
- q = Probability of Failure in Each Trial.
The Binomial Distribution Formula which is used to calculate the probability, for a random variable X = 0, 1, 2, 3,....,n is given as
P(X = r) = nCrprqn-r, r = 0, 1, 2, 3....
Where,
- n = Total number of trials
- r = Number of successes
- p = Probability of success
- q = Probability of failure (q = 1 - p)
Bernoulli Trials in Binomial Distribution
Bernoulli Trial is a trial that gives results of dichotomous nature i.e. result in yes or no, head or tail, even or odd. It means it gives two types of outcomes out of which one favors the event while the other doesn't. A random experiment is called Bernoulli Trial if it satisfies the following conditions:
- Trials are finite in number
- Trials are independent of each other
- Each trial has only two possible outcomes
- The probability of success and failure in each trial is the same.
Binomial Random Variable
A Binomial Random Variable can be defined by two possible outcomes such as “success” and binomial “failure”. For instance, consider rolling a fair six-sided die and recording the value of the face. The binomial distribution formula can be put into use to calculate the probability of success for binomial distributions. Often it states “plugin” the numbers to the formula and calculates the requisite values.
The binomial distribution is based on the following characteristics:
- Experiment contains n identical trials.
- Each trial results in one of the two outcomes either success or failure.
- The probability of success, denoted p, remains the same from trial to trial.
- All the n trials are independent.
Example: A fair coin is flipped 20 times;
X represents the number of heads
X is a binomial random variable with n = 20 which is the total number of trials and p = 1/2 is the probability of getting head in each trial.
The value of X represents the number of trials in which you succeed in getting head.
Binomial Distribution Calculation
Binomial Distribution in statistics is used to compute the probability of likelihood of an event using the above formula. To calculate the probability using binomial distribution we need to follow the following steps:
- Step 1: Find the number of trials and assign it as 'n'
- Step 2: Find the probability of success in each trial and assign it as 'p'
- Step 3: Find the probability of failure and assign it as q where q = 1-p
- Step 4: Find the random variable X = r for which we have to calculate the binomial distribution
- Step 5: Calculate the probability of Binomial Distribution for X = r using the Binomial Distribution Formula.
The use of the above steps has been illustrated using an example below:
Binomial Distribution Examples
- Finding the probability of getting exactly 6 heads when a fair coin is flipped 10 times.
- Finding the probability of exactly 3 bulbs being defective when a batch of 100 bulbs is tested and each bulb has a 2% chance of being defective.
- To find the Probability of exactly 7 patients responding positively to the treatment when the drug is tested on 8 patients and has a 90% success rate.
Let's say we toss a coin twice, and getting head is a success we have to calculate the probability of success and failure then, in this case, we will calculate the probability distribution as follows:
In each trial getting a head that is a success, its probability is given as:
- p = 1/2
- n = 2 as we throw a coin twice
- r = 0 for no success, r = 1 for getting head once and r = 2 for getting head twice
Probability of failure q = 1 - p = 1 - 1/2 = 1/2.
P(Getting 1 head) = P(X = 1) = ncrprqn-r = 2c1 (1/2)1(1/2)1 = 2 ⨯ 1/2 ⨯ 1/2 = 1/2
P(Getting 2 heads) = P(X = 2) = 2c2(1/2)2(1/2)0 = 1/4
P(Getting 0 heads) = P(X = 0) = 2c0(1/2)0(1/2)2 = 1/4
Random Variable (X = r) | P(X = r) |
|---|
X = 0 (Getting 0 Head) | 1/4 |
X = 1 (Getting 1 Head) | 1/2 |
X = 2 (Getting 2 Head) | 1/4 |
As of now, we know that Binomial Distribution is calculated for the Random Variables obtained in Bernoulli Trials. Hence, we should understand these terms.
Binomial Distribution Table
The binomial distribution for a situation when getting 6 is a success on throwing two dies is discussed in this section. First of all, we see that it is a Bernoulli Trial as getting 6 is the only success, and getting any different is a failure. Now we can get six on both die in a trial or six on only one of the die in a trial and getting no six on both die. Hence, the random variable for which we have to find the probability takes the value X = r = 0, 1, 2.
The Binomial Distribution Table for getting 6 as success is plotted below:
Random Variable (X = r) | P(X = r) |
|---|
X = 0 (Getting no 6) | 25/36 |
X = 1 (Getting one 6) | 10/36 |
X = 2 (Getting two 6) | 1/36 |
We see that sum of all the probabilities 25/36 + 10/36 + 1/36 = 1.
Binomial Distribution Graph
Binomial Distribution Graph is plotted for X and P(X). We will plot a Binomial Distribution Graph for tossing a coin twice where getting the head is a success. If we toss a coin twice, the possible outcomes are {HH, HT, TH, TT}. The binomial distribution Table for this is given below:
X (Random Variable) | P(X) |
|---|
X = 0 (Getting no head) | P(X = 0) = 1/4 = 0.25 |
X = 1 (Getting 1 head) | P(X = 1) = 2/4 = 1/2 = 0.5 |
X = 2 (Getting two heads) | P(X = 2) = 1/4 = 0.25 |
Binomial Distribution Graph for the above table is given below:
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Binomial Distribution in Statistics
Measures of central tendency, specifically the mean, provide insights into the distribution's central or typical value for the number of successes in a series of independent trials. For a binomial distribution defined by parameters n (number of trials) and p (probability of success on each trial), the measures of central tendency are characterized as follows:
- Binomial Distribution Mean
- Binomial Distribution Variance
- Binomial Distribution Standard Deviation
Measure of Central Tendency for Binomial Distribution
The formulas for Mean, Variance, and Standard Deviation of Binomial Distribution are listed below:
Binomial Distribution Mean
The Mean of Binomial Distribution is the measurement of average success that would be obtained in the 'n' number of trials. The Mean of Binomial Distribution is also called Binomial Distribution Expectation. The formula for Binomial Distribution Expectation is given as:
μ = n.p
where,
- μ is the Mean or Expectation
- n is the Total Number of Trials
- p is the Probability of Success in Each Trial
Read more about, Expected Value or Expectation
Example: If we toss a coin 20 times and getting head is the success then what is the mean of success?
Solution:
Total Number of Trials n = 20
Probability of getting head in each trial, p = 1/2 = 0.5
Mean = n.p = 20 ⨯ 0.5
It means on average we would head 10 times on tossing a coin 20 times.
Binomial Distribution Variance
Variance of Binomial Distribution tells about the dispersion or spread of the distribution. It is given by the product of the number of trials, probability of success, and probability of failure. The formula for Variance is given as follows:
σ2 = n.p.q
where
- σ2 is Variance
- n is the Total Number of Trials
- p is the Probability of Success in Each Trial
- q is the Probability of Failure in Each Trial
Example: If we toss a coin 20 times and getting head is the success then what is the variance of the distribution?
Solution:
We have, n = 20
Probability of Success in each trial (p) = 0.5
Probability of Failure in each trial (q) = 0.5
Variance of the Binomial Distribution, σ = n.p.q = (20 ⨯ 0.5 ⨯ 0.5) = 5
Binomial Distribution Standard Deviation
Standard Deviation of Binomial Distribution tells about the deviation of the data from the mean. Mathematically, Standard Deviation is the square root of the variance. The formula for the Standard Deviation of Binomial Distribution is given as
σ = √n.p.q
where,
- σ is the Standard Deviation
- n is the Total Number of Trials
- p is the Probability of Success in Each Trial
- q is the Probability of Failure in Each Trial
Example: If we toss a coin 20 times and getting head is the success then what is the standard deviation?
Solution:
We have, n = 20
Probability of Success in each trial (p) = 0.5
Probability of Failure in each trial (q) = 0.5
Standard Deviation of the Binomial Distribution, σ = √n.p.q
⇒ σ = √(20 ⨯ 0.5 ⨯ 0.5)
⇒ σ = √5 = 2.23
Binomial Distribution Properties
Properties of Binomial Distribution are mentioned below:
- There are only two possible outcomes: success or failure, yes or no, true or false.
- There is a finite number of trials given as 'n'.
- The probability of success and failure in each trial is the same.
- Only Success is calculated out of all trials.
- Each trial is independent of any other trial.
Binomial Distribution Applications
Binomial Distribution is used where we have only two possible outcomes. Let's see some of the areas where Binomial Distribution can be used.
- To find the number of male and female students in an institute.
- To find the likeability of something in Yes or No.
- To find defective or good products manufactured in a factor.
- To find positive and negative reviews on a product.
- Votes are collected in the form of 0 or 1.
Binomial Distribution vs Normal Distribution
Binomial Distribution differs from the Normal Distribution in many aspects. The key differences and characteristics of the Binomial and Normal distributions are highlighted in the following table:
| Aspect | Binomial Distribution | Normal Distribution |
|---|
| Type | Discrete probability distribution | Continuous probability distribution |
| Outcomes | Two possible outcomes per trial (success or failure) | Infinite possible outcomes within a continuous range |
| Parameters | n (number of trials), p (probability of success) | μ (mean), σ (standard deviation) |
| Shape | Varies depending on n and p; typically skewed unless p=0.5 and n is large | Bell-shaped curve (symmetric) |
| Support | x can take integer values from 0 to n | x can take any real number (from −∞ to +∞) |
| Mean | μ = np | μ |
| Variance | 𝝈2 = np(1 -p) | 𝝈2 |
| Applicability | Used for modeling the number of successes in a fixed number of independent trials | Used for modeling continuous data that cluster around a mean |
| Examples | Flipping coins, quality control (defective items) | Heights of people, test scores, measurement errors |
| Approximation | Approximates Normal distribution for large n and p not too close to 0 or 1 | Considered the limit of the Binomial Distribution as n becomes large and p is near 0.5 |
People Also Read:
Binomial Distribution in Probability Examples
Example 1: A die is thrown 6 times and if getting an even number is a success what is the probability of getting
(i) 4 Successes
(ii) No success
Solution:
Given: n = 6, p = 3/6 = 1/2, and q = 1 - 1/2 = 1/2
P(X = r) = nCrprqn-r
(i) P(X = 4) = 6C4(1/2)4(1/2)2 = 15/64
(ii) P(X = 0) = 6C0(1/2)0(1/2)6 = 1/64
Example 2: A coin is tossed 4 times what is the probability of getting at least 2 heads?
Solution:
Given: n = 4
Probability of getting head in each trial, p = 1/2 ⇒ q = 1 - 1/2 = 1/2
P(X = r) = 4Cr(1/2)r(1/2)4-r
⇒ P(X = r) = 4Cr(1/2)4 {Using the laws of Exaponents}
And we know, Probability of getting at least 2 heads = P(X ≥ 2)
⇒ Probability of getting at least 2 heads = P(X = 2) + P(X = 3) + P(X = 4)
⇒ Probability of getting at least 2 heads = 4C2(1/2)4 + 4C3(1/2)4 + 4C4(1/2)4
⇒ Probability of getting at least 2 heads = (4C2 + 4C3 + 4C4)(1/2)4
⇒ Probability of getting at least 2 heads = 11(1/2)4 = 11/16
Example 3: A pair of dice is thrown 6 times and getting sum 5 is a success then what is the probability of getting (i) no success (ii) two success (iii) at most two success
Solution:
Given: n = 6
5 can be obtained in 4 ways (1, 4) (4, 1) (2, 3) (3, 2)
Probability of getting the sum 5 in each trial, p = 4/36 = 1/9
Probability of not getting sum 5 = 1 - 1/9 = 8/9
(i) Probability of getting no success, P(X = 0) = 6C0(1/9)0(8/9)6 = (8/9)6
(ii) Probability of getting two success, P(X = 2) = 6C2(1/9)2(8/9)4 = 15(84/96)
(iii) Probability of getting at most two successes, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
⇒ P(X ≤ 2) = (8/9)6 + 6(85/96) + 15(84/96)
Practice Problems on Binomial Distribution in Probability
1. A box has 5 red, 7 black,? and 8 white balls. If three balls are drawn one by one with replacement what is the probability that all,
i) all are white
ii) all are red
iii) all are black
2. What is the probability distribution of the number of tails when three coins are tossed together?
3. A die is thrown three times what is the probability distribution of getting six?
4. A coin is tossed 4 times then what is the probability distribution of getting head.
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Integration by Partial FractionsIntegration by Partial Fractions is one of the methods of integration, which is used to find the integral of the rational functions. In Partial Fraction decomposition, an improper-looking rational function is decomposed into the sum of various proper rational functions.If f(x) and g(x) are polynomia
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Integration by PartsIntegration by Parts or Partial Integration, is a technique used in calculus to evaluate the integral of a product of two functions. The formula for partial integration is given by:∫ u dv = uv - ∫ v duWhere u and v are differentiable functions of x. This formula allows us to simplify the integral of
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Integration of Trigonometric FunctionsIntegration is the process of summing up small values of a function in the region of limits. It is just the opposite to differentiation. Integration is also known as anti-derivative. We have explained the Integration of Trigonometric Functions in this article below.Below is an example of the Integra
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Functions Defined by IntegralsWhile thinking about functions, we always imagine that a function is a mathematical machine that gives us an output for any input we give. It is usually thought of in terms of mathematical expressions like squares, exponential and trigonometric function, etc. It is also possible to define the functi
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Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite
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Computing Definite IntegralsIntegrals are a very important part of the calculus. They allow us to calculate the anti-derivatives, that is given a function's derivative, integrals give the function as output. Other important applications of integrals include calculating the area under the curve, the volume enclosed by a surface
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Fundamental Theorem of Calculus | Part 1, Part 2Fundamental Theorem of Calculus is the basic theorem that is widely used for defining a relation between integrating a function of differentiating a function. The fundamental theorem of calculus is widely useful for solving various differential and integral problems and making the solution easy for
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Finding Derivative with Fundamental Theorem of CalculusIntegrals are the reverse process of differentiation. They are also called anti-derivatives and are used to find the areas and volumes of the arbitrary shapes for which there are no formulas available to us. Indefinite integrals simply calculate the anti-derivative of the function, while the definit
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Evaluating Definite IntegralsIntegration, as the name suggests is used to integrate something. In mathematics, integration is the method used to integrate functions. The other word for integration can be summation as it is used, to sum up, the entire function or in a graphical way, used to find the area under the curve function
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Properties of Definite IntegralsProperties of Definite Integrals: An integral that has a limit is known as a definite integral. It has an upper limit and a lower limit. It is represented as \int_{a}^{b}f(x) = F(b) − F(a)There are many properties regarding definite integral. We will discuss each property one by one with proof.Defin
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Definite Integrals of Piecewise FunctionsImagine a graph with a function drawn on it, it can be a straight line or a curve, or anything as long as it is a function. Now, this is just one function on the graph. Can 2 functions simultaneously occur on the graph? Imagine two functions simultaneously occurring on the graph, say, a straight lin
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Improper IntegralsImproper integrals are definite integrals where one or both of the boundaries are at infinity or where the Integrand has a vertical asymptote in the interval of integration. Computing the area up to infinity seems like an intractable problem, but through some clever manipulation, such problems can b
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Riemann SumsRiemann Sum is a certain kind of approximation of an integral by a finite sum. A Riemann sum is the sum of rectangles or trapezoids that approximate vertical slices of the area in question. German mathematician Bernhard Riemann developed the concept of Riemann Sums. In this article, we will look int
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Riemann Sums in Summation NotationRiemann sums allow us to calculate the area under the curve for any arbitrary function. These formulations help us define the definite integral. The basic idea behind these sums is to divide the area that is supposed to be calculated into small rectangles and calculate the sum of their areas. These
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Trapezoidal RuleThe Trapezoidal Rule is a fundamental method in numerical integration used to approximate the value of a definite integral of the form b∫a f(x) dx. It estimates the area under the curve y = f(x) by dividing the interval [a, b] into smaller subintervals and approximating the region under the curve as
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Definite Integral as the Limit of a Riemann SumDefinite integrals are an important part of calculus. They are used to calculate the areas, volumes, etc of arbitrary shapes for which formulas are not defined. Analytically they are just indefinite integrals with limits on top of them, but graphically they represent the area under the curve. The li
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Antiderivative: Integration as Inverse Process of DifferentiationAn antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that:d/dx[F(x)] = f(x)In other words, F(x) is a function whose derivative is f(x).Antiderivatives include a family of functions t
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Indefinite IntegralsIntegrals are also known as anti-derivatives as integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and are required to calculate the function from the derivative. This process is called integration or anti-different
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Particular Solutions to Differential EquationsIndefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
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Integration by U-substitutionFinding integrals is basically a reverse differentiation process. That is why integrals are also called anti-derivatives. Often the functions are straightforward and standard functions that can be integrated easily. It is easier to solve the combination of these functions using the properties of ind
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Reverse Chain RuleIntegrals are an important part of the theory of calculus. They are very useful in calculating the areas and volumes for arbitrarily complex functions, which otherwise are very hard to compute and are often bad approximations of the area or the volume enclosed by the function. Integrals are the reve
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Partial Fraction ExpansionIf f(x) is a function that is required to be integrated, f(x) is called the Integrand, and the integration of the function without any limits or boundaries is known as the Indefinite Integration. Indefinite integration has its own formulae to make the process of integration easier. However, sometime
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Trigonometric Substitution: Method, Formula and Solved ExamplesTrigonometric substitution is a process in which the substitution of a trigonometric function into another expression takes place. It is used to evaluate integrals or it is a method for finding antiderivatives of functions that contain square roots of quadratic expressions or rational powers of the
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Chapter 8: Applications of Integrals
Area under Simple CurvesWe know how to calculate the areas of some standard curves like rectangles, squares, trapezium, etc. There are formulas for areas of each of these figures, but in real life, these figures are not always perfect. Sometimes it may happen that we have a figure that looks like a square but is not actual
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Area Between Two Curves: Formula, Definition and ExamplesArea Between Two Curves in Calculus is one of the applications of Integration. It helps us calculate the area bounded between two or more curves using the integration. As we know Integration in calculus is defined as the continuous summation of very small units. The topic "Area Between Two Curves" h
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Area between Polar CurvesCoordinate systems allow the mathematical formulation of the position and behavior of a body in space. These systems are used almost everywhere in real life. Usually, the rectangular Cartesian coordinate system is seen, but there is another type of coordinate system which is useful for certain kinds
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Area as Definite IntegralIntegrals are an integral part of calculus. They represent summation, for functions which are not as straightforward as standard functions, integrals help us to calculate the sum and their areas and give us the flexibility to work with any type of function we want to work with. The areas for the sta
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Chapter 9: Differential Equations
Differential EquationsA differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur
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Particular Solutions to Differential EquationsIndefinite integrals are the reverse of the differentiation process. Given a function f(x) and it's derivative f'(x), they help us in calculating the function f(x) from f'(x). These are used almost everywhere in calculus and are thus called the backbone of the field of calculus. Geometrically speaki
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Homogeneous Differential EquationsHomogeneous Differential Equations are differential equations with homogenous functions. They are equations containing a differentiation operator, a function, and a set of variables. The general form of the homogeneous differential equation is f(x, y).dy + g(x, y).dx = 0, where f(x, y) and h(x, y) i
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Separable Differential EquationsSeparable differential equations are a special type of ordinary differential equation (ODE) that can be solved by separating the variables and integrating each side separately. Any differential equation that can be written in form of y' = f(x).g(y), is called a separable differential equation. Separ
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Exact Equations and Integrating FactorsDifferential Equations are used to describe a lot of physical phenomena. They help us to observe something happening in real life and put it in a mathematical form. At this level, we are mostly concerned with linear and first-order differential equations. A differential equation in “y†is linear if
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Implicit DifferentiationImplicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s
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Implicit differentiation - Advanced ExamplesIn the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation. Table of Content Implicit DifferentiationMethod to solveImplicit differentiation Formula Solve
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Advanced DifferentiationDerivatives are used to measure the rate of change of any quantity. This process is called differentiation. It can be considered as a building block of the theory of calculus. Geometrically speaking, the derivative of any function at a particular point gives the slope of the tangent at that point of
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Disguised Derivatives - Advanced differentiation | Class 12 MathsThe dictionary meaning of “disguise†is “unrecognizableâ€. Disguised derivative means “unrecognized derivativeâ€. In this type of problem, the definition of derivative is hidden in the form of a limit. At a glance, the problem seems to be solvable using limit properties but it is much easier to solve
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Derivative of Inverse Trigonometric FunctionsDerivative of Inverse Trigonometric Function refers to the rate of change in Inverse Trigonometric Functions. We know that the derivative of a function is the rate of change in a function with respect to the independent variable. Before learning this, one should know the formulas of differentiation
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Logarithmic DifferentiationMethod of finding a function's derivative by first taking the logarithm and then differentiating is called logarithmic differentiation. This method is specially used when the function is type y = f(x)g(x). In this type of problem where y is a composite function, we first need to take a logarithm, ma
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