Binomial Distribution in Probability
Last Updated :
26 Jul, 2025
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.
Binomial Distribution GraphConditions for Binomial Distribution
The Binomial distribution can be used in scenarios where the following conditions are satisfied:
- Fixed Number of Trials: There is 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.
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) = nCr pr (1-p) n-r, r = 0, 1, 2, 3....
Where,
- n = Total number of trials
- r = Number of successes
- p = Probability of success
Binomial Random Variable
A binomial random variable X counts the number of "successes" in n independent trials, with two outcomes in each trial: success (with probability p) or failure (with probability 1−p) and constant probability p across all trials.
Example:
A fair coin is flipped 20 times;
Success: "Heads" (p=0.5).
Random variable X: Number of heads observed in 20 flips.
Distribution: X∼Binomial (n=20,p=0.5).
Probability of Getting exacly 10 heads is given by, (r=10)
P(X=10)=^{10}C_{20} (0.5)^{10}(0.5)^{10} ≈ 0.176(17.6\%)
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-1 pr(1-p) n
Where,
- n = Total Number of Trials.
- r = Number of Trials in which we get the first success.
- p = Probability of Success in Each Trial.
- (1-p) = Probability of Failure in Each Trial.
Bernoulli Trials in Binomial Distribution
Bernoulli Trial are a sequence of independent experiments, where each experiment (or trial) results in exactly two possible outcomes:
- Success (with probability p)
- Failure (with probability 1−p)
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.
The binomial distribution models the number of successes in a fixed number of Bernoulli trials.
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 Visualization
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) {TT} | P(X = 0) = 1/4 = 0.25 |
X = 1 (Getting 1 head) {HT, TH} | P(X = 1) = 2/4 = 1/2 = 0.5 |
X = 2 (Getting two heads) {HH} | 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(Expected Value or 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
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 get head 10 times on tossing a coin 20 times.
Binomial Distribution Variance
Varianceof 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
- σ2is 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 = \sqrt {n\cdot p \cdot 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 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:
| Binomial Distribution | Normal Distribution |
|---|
| Discrete probability distribution | Continuous probability distribution |
| Two possible outcomes per trial (success or failure) | Infinite possible outcomes within a continuous range |
| Varies depending on n and p; typically skewed unless p=0.5 and n is large | Bell-shaped curve (symmetric) |
| x can take integer values from 0 to n | x can take any real number (from −∞ to +∞) |
μ = np
(n = number of trials, p = success probability) | μ (mean; center of the curve, given directly) |
𝝈2 = np(1 -p) | 𝝈2 (variance; spread, given directly) |
| Used for modeling the number of successes in a fixed number of independent trials | Used for modeling continuous data that cluster around a mean |
| Flipping coins, quality control (defective items) | Heights of people, test scores, measurement errors |
| 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 |
Related Articles
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 ofdice 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.
Answers:
1. (i). 8/125 (ii). 1/64 (iii). 343/8000
2.
No. of tails | Probability |
|---|
0 | 1/8 |
|---|
1 | 3/8 |
|---|
2 | 3/8 |
|---|
3 | 1/8 |
|---|
3.
No. of sixes | Probability |
|---|
0 | 125/216 |
|---|
1 | 75/216 |
|---|
2 | 15/216 |
|---|
3 | 1/216 |
|---|
4.
No. of heads | Probability |
|---|
0 | 1/16 |
|---|
1 | 1/4 |
|---|
2 | 3/8 |
|---|
3 | 1/4 |
|---|
4 | 1/16 |
|---|
Binomial Distribution in Probability | Formula, Definition & Examples
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